• 


O'REILLY  ESTATE  BUILDING,  ST.  LOUIS,  MO. 

A.  B.  Groves,  Architect  Murch  Bros.  Construction  Co.,  Contractors 

G.  S.  Bergendahl,  M.  AM.,  Soc.  C.  E.,  Engineer,  St.  Louis  Representative,  Mushroom  System 


THE  THEORY  OF  THE  FLEXURE 
AND  STRENGTH 

OF 

RECTANGULAR    FLAT 
PLATES 

APPLIED    TO 

REINFORCED   CONCRETE 
FLOOR   SLABS 


BY 


HENRY  T.  IJDDY,  C.  E.,  Ph.  D.,  Sc.  D. 

PROFESSOR  OF  MATHEMATICS  AND   MECHANICS,  COLLEGE  OF  ENGINEERING, 

AND  DEAN  OF  THE  GRADUATE  SCHOOL,  EMERITUS,  UNIVERSITY  OF  MINNESOTA. 

CHAIRMAN  OF  BOARD  OF  EXAMINERS  FOR  THE  MUSHROOM  SYSTEM 

APPOINTED  BY  THE  COMMISSIONER  OF  BUILDINGS 

CHICAGO,  JUNE  15,  1912 


PRICE  $5.00 


ROGERS  &  COMPANY,  MINNEAPOLIS 
1913 


COPYRIGHT  1913 
BY 

HENRY  T.EDDY 

ALL  RIGHTS  RESERVED 


PREFACE 

In  reviewing  the  history  of  every  type  of  structural  work,  we  find 
the  designing  engineer  influenced  in  his  first  attempts  at  any  new 
type  of  structure  by  his  knowledge  of  the  practical  forms  of  con- 
struction with  which  he  is  already  familiar.  In  fact  until  very 
recent  times  precedent  was  the  engineer's  sole  guide.  It  was  to  be 
expected  therefore  that  the  pioneers  in  concrete-steel  construction 
should  follow  and  closely  imitate  timber  and  structural  steel  con- 
struction. In  following  this  type,  the  idea  has  been  to  build  up  the 
structure  as  a  whole  by  assembling  and  joining  together  a  number 
of  independent  elements  or  units:  whereas  concrete,  with  or  without 
reinforcement,  is  a  kind  of  material  that  is  best  suited  by  its  nature 
to  construction  in  monolithic  form.  But  when  the  attempt  has  been 
made  to  treat  such  structures  theoretically,  preconceived  ideas 
have  led  to  the  effort  to  treat  them  by  analysing  them  into  separate 
members  and  computing  the  strength  of  these  arbitrarily  selected 
units,  assumed  to  act  independently  as  they  do  in  steel  structures. 
Such  treatment  has  led  to  errors  of  as  much  as  two  or  three  hundred 
percent  in  the  computation  of  slabs  with  two  way  reinforcement 
supported  on  four  sides,  and  to  errors  of  four  hundred  percent  in 
case  of  continuous  flat  slab  construction  such  as  occurs  in  the  mush- 
room system. 

When  we  consider  the  fact  that  fire  losses  in  Canada  and  the 
United  States  amount  each  year  to  half  a  billion  dollars,  and  that 
the  question  of  commercial  economy  determines  whether  buildings 
shall  be  built  of  fireproof  and  incombustible  materials  such  as  rein- 
forced concrete,  or  of  inflammable  materials  such  as  are  used  in  timber 
construction,  it  is  at  once  evident  how  important  it  is  to  the  general 
public  to  be  able  to  determine  on  theoretically  correct  principles 
whether  safe  fireproof  buildings  can  be  built  at  approximately  the 
same  or  less  cost  than  combustible  ones.  In  case  of  any  uncertainty 
on  this  question,  the  designer  is  compelled  for  safety  to  employ 
materials  in  such  lavish  amounts  as  to  render  cost  prohibitive. 

The  failure  of  engineers  and  mathematicians  generally  to  apply 
the  mathematical  theory  of  elasticity  to  the  new  material  concrete- 
steel,  has  led  to  considerable  controversy  between  practical  con- 
structors of  experience,  and  theoretical  engineers  without  such 
experience. 


IV  PREFACE 

Marsh  in  his  treatise  on  Reinforced  Concrete,  edition  of  1905, 
Part  V,  p.  209,  makes  the  following  remarks  upon  this  subject: — 

"When  properly  combined  with  metal,  concrete  appears  to  gain 
properties  which  do  not  exist  in  the  material  when  by  itself,  and 
although  much  has  been  done  by  various  experimenters  in  recent 
years  to  increase  our  knowledge  on  the  subject  of  the  elastic  be- 
haviour of  reinforced  concrete,  we  are  still  very  far  from  having 
a  true  perception  of  the  characteristics  of  the  composite  material. 

It  may  be  that  we  are  wrong  from  the  commencement  in 
attempting  to  treat  it  after  the  manner  of  structural  ironwork,  and 
that  although  the  proper  allowances  for  the  elastic  properties  of  the 
dual  material  is  an  advancement  on  the  empirical  formulae  at  first 
employed,  and  used  by  many  constructors  at  the  present  time,  yet 
we  may  be  entirely  wrong  in  our  method  of  treatment. 

The  molecular  theory,  i.  e.  the  prevention  of  molecular  defor- 
mation by  supplying  resistances  of  the  reverse  kind  to  the  stresses 
on  small  particles,  may  prove  to  be  the  true  method  of  treatment  for 
a  composite  material  such  as  concrete  and  metal.  This  theory  is 
the  basis  of  the  Cottacin  construction  which  certainly  produces  good 
results  and  very  light  structures,  and  M.  Considere's  latest  researches 
on  the  subject  of  hooped  concrete  are  somewhat  on  these  lines. » 

In  this  statement,  Marsh  undoubtedly  has  in  mind  the  great 
discrepancy  between  the  results  of  tests  and  of  computations  of 
multiple  way  reinforcement. 

The  empirical  formulas  of  Hennebique,  a  pioneer  in  this  field 
and  one  of  its  most  extensive  investigators,  give  numerical  results 
at  variance  with  the  usual  published  theories  for  two  way  reinforce- 
ment, while  the  empirical  formulas  published  by  Turner  in  1908 
exhibit  an  equally  radical  divergence. 

It  has  remained  apparently  for  Dr.  Eddy  to  discover  the  reason 
for  this  great  discrepancy  by  a  rigid  application  of  the  mathematical 
theory  of  elasticity  to  the  problem  presented  by  multiple  way  rein- 
forcement. 

The  clearing  up  of  the  mathematical  difficulties  with  which  the 
theoretical  engineer  has  heretofore  struggled  in  dealing  with 
reinforced  concrete,  will  lead  to  its  more  general  adoption  by  the 
elimination  of  uncertainty  hi  design;  and  will  lead  to  the  adoption 
of  those  types  which  are  safest  to  erect,  and  those  which  possess  a 
degree  of  toughness,  due  to  their  monolithic  construction,  which  is 
lacking  in  types  that  merely  imitate  older  forms  of  timber  and  steel 
construction. 

The  record  of  the  Mushroom  system  in  the  successful  construc- 
tion of  between  one  and  two  thousand  buildings,  without  accident 
to  the  workmen,  and  without  failure  to  make  good  the  guarantee 


PREFACE  V 

of  test  capacity,  can  be  accounted  for  only  on  the  ground  that 
scientifically  designed  multiple  way  reinforcement  is  inherently  safer 
to  erect,  and  more  reliable  at  all  stages  of  construction  than  other 
types.  The  theory  given  in  detail  which  accounts  for  its  economy 
and  safety,  should,  we  believe  be  of  interest  to  the  profession  at 
large,  and  should  prove  of  value  to  the  practicing  engineer  in  check- 
ing designs  for  his  clients. 

The  endorsement  of  this  theory  by  the  undersigned  does  not 
constitute  a  license  to  use  the  patented  type,  though  the  extension 
of  the  theory  to  older  types  which  are  not  patented  will  undoubtedly 
lead  to  the  closest  competition  with  it  along  legitimate  lines.  Any 
loss  to  the  patentee  arising  from  such  competition,  will,  in  the  writer's 
judgment,  be  more  than  counterbalanced  from  a  commercial  stand- 
point, by  the  increased  safety  and  corresponding  popularity  of 
reinforced  concrete  as  a  material  of  construction. 

The  invention  covered  by  the  claims  of  the  broad  patent 
just  mentioned  includes  much  more  than  merely  the  standard  mush- 
room system.  This  is,  however,  the  one  form  of  all  others 
of  the  patent,  which  the  writer  prefers  by  reason  of  certain  practi- 
cal advantages  which  it  possesses.  Its  arrangement  of  parts  whereby 
a  continous  multiple  way  reinforced  flat  slab  is  supported  by  a 
large  cantilever  column  head  integral  with  the  column  and  em- 
bedded in  the  slab  so  as  to  resist  tensile  stresses  both  radial 
and  circumferential  in  zones  near  the  top  of  the  slab  over 
and  around  the  columns,  and  in  the  zones  near  the  bottom  of  the 
slab  toward  the  center  of  the  panels,  will  be  fully  discussed  in  succeed- 
ing pages.  This  discussion,  which  deals  with  the  mushroom  sys- 
tem primarily,  is  intended  as  an  advance  chapter  or  two  of  a  more 
comprehensive  treatise  on  Concrete-Steel  Construction,  in  which 
it  is  intended  to  treat  somewhat  fully  concrete  columns,  beam  and 
slab  construction,  wall  panels,  etc.,  as  well  as  flat  slabs,  and  to 
introduce  the  results  of  experimental  work  now  under  way  to 
determine  the  value  of  Poisson's  ratio  for  different  combinations 
of  steel  and  concrete. 

This  treatise  will  then  represent  the  joint  efforts  of  a  pro- 
fessional mathematician  accustomed  to  treating  these  problems, 
and  a  professional  builder  and  designer  of  reinforced  concrete  with 
many  years  of  practical  experience  behind  him. 

The  price  charged  for  this  booklet  will  be  credited  in  return 
for  it,  on  the  larger  treatise  which  the  authors  intend  to  complete 
as  soon  as  the  magnitude  of  the  task  will  permit. 

C.  A.  P.  TURNER. 


CONTENTS 


Section  Page 

1.  Flat  Slab  Floors.     The  Mushroom  System ?.  .  1-4 

2.  Notation v  .  5-7 

3.  True  and  Apparent  Bending  Moments 8- 10 

4.  Poisson's  Ratio 11-12 

5.  General  Differential  Equation  of  Moments 13-15 

6.  General  Differential  Equation  of  Deflections 16-17 

7.  Solution  of  the  Differential  Equation  in   Case   of    Uni- 

form Slab  Supported  on  Columns 17-20 

8.  Solution  for  Side  Belts 21-24 

9.  Practical  Formulas  for  Stresses  in  Side  Belts 24-30 

10.  Practical  Formulas  for  Stresses  in  Column  Heads 31-35 

11.  Practical  Formulas  for  Stresses  in  the  Middle  Area  of 

Panel 36-40 

12.  Deflections  at  Mid  Span  of  the  Side  and  Diagonal  Belts.  41-43 

13.  Proportionate  Deflections  at  Mid  Span  and    Center  of 

Panel 43-46 

14.  Radial  and  Ring  Rods,  and  Shear  around  Cap 47-55 

15.  Standard  Mushroom  System  and  other  Systems 55-61 

16.  Specimen  Computation  of  Slab 62-70 

17.  Comparative  Test  to  Destruction  of  Two  Slabs 71-93 

18.  Suggestions  as  to  Construction,  Finish,  Etc 94-100 

Appendix.     Specifications  for  Floors 101-104 


VII 


TISCHERS  CREEK  BRIDGE,  DULUTH,  MINN. 

Spans  are  26  feet  longtitudinally 
This  type  is  built  with  spans  up  to  50'0" 


View  of  Reinforcement  in  Place 

TISCHERS  CREEK  BRIDGE,  DULUTH,  MINN. 
Designed  by  C.  A.  P.  Turner  Geo.  H.  Lounsbury,  Contractor 


FLAT  SLAB  FLOORS 

1 .  The  superiority  of  flat  slab  floors  supported  directly  on  columns, 
over  other  forms  of  construction  when  looked  at  from  the  stand- 
point of  lower  cost,  better  lighting,  greater  neatness  of  appearance, 
and  increased  safety  and  rapidity  of  construction,  is  so  generally,  or 
rather  so  universally  conceded  as  to  render  any  reliable  information 
relative  to  the  scientific  computation  of  stresses  in  this  type  of  con- 
struction of  great  interest.  Heidenreich,  in  his  Engineer's  Pocket 
Book  on  Reinforced  Concrete,  page  89,  classifies  this  type  as  floors 
without  beams  and  girders — " Mushroom  System." 

Since  "mushroom,"  as  applied  to  concrete,  is  an  arbitrary  or 
fanciful  term,  and  indeed,  almost  a  contradictory  one,  a  word  of 
explanation  as  to  its  origin  may  be  of  interest.  The  term  was 
originated  by  C.  A.  P.  Turner,  of  Minneapolis,  and  applied  to  his 
flat  plate  construction,  more  particularly  because  of  the  fancied 
resemblance  to  the  mushroom,  of  the  column  and  column  head 
reinforcement  of  that  particular  form  of  his  flat  plate  construction 
which  he  seemed  to  prefer  by  reason  of  certain  practical  advantages. 
Another  fancied  resemblance  is  the  rapidity  of  erection,  comparable 
to  the  over-night  growth  of  the  mushroom.  Here  the  resemblance 
ceases,  since  the  construction,  once  erected,  is  enduring  and  per- 
manent. 

The  Mushroom  System  is  a  continuous  flat  plate  of  concrete 
supported  directly  on  columns,  and  reinforced  in  such  a  maner  that 
circular  and  radial  tensile  stresses  concentric  with  the  column  are 
provided  for  by  metal  reinforcement  in  the  tension  zone  above  the 
columns,  and  similar  provision  is  made  for  tensile  stresses  in  the 
lower  portion  of  the  slab  concentric  with  the  center  of  the  panel, 
diagonally  between  the  columns.  Since  all  forces  in  a  plane  may  be 
resolved  into  equivalent  components  along  any  pair  of  axes  at  right 
angles  to  each  other,  it  is  possible  to  provide  reinforcement  to  resist 
any  horizontal  tensile  stresses  in  the  slab  by  various  arrangements 
of  intersecting  belts  of  rods  at  zones  where  these  stresses  occur. 
All  arrangements  of  this  kind  are  by  no  means  equally  effective. 
A  system  of  wide  reinforcing  belts  from  column  to  column  com- 
bined with  a  system  of  radial  and  ring  rods  to  constitute  a  large, 
substantial  cantilever  mushroom  head  at  the  top  of  each  column 


e-'    to  -o,  tc-c 


STORY  COL 


NO 


Fig.  1.  Vertical  Section  of  Standard  Mushroom  Head  showing  posi- 
tion of  Radial  and  Ring  Rods,  and  Slab  Rods,  Vertical  and  Hori- 
zontal Sections  of  Spirally  Hooped  Column,  with  Plain  Bar  Hoop 
Collar  Band,  Vertical  Reinforcing  Rods  and  Elbow  Rods. 


Fig.   2.     Plan  of  Reinforcement   in  Standard   Mushroom   System. 
Radial  and  Ring  Rods,  Collar  Band  and  Slab  Rods.     Diameter  of 


STANDARD    MUSHROOM    SYSTEM  3 

provides  a  very  effective  and  economical  arrangement  for  controlling 
the  distribution  of  the  stresses  in  the  slab,  and  furnishes  the  resis- 
tance necessary  to  support  these  stresses  by  placing  the  steel  where  it 
is  most  needed.  It  not  only  has  the  same  kind  of  advantage  that  the 
continuous  cantilever  beam  -has  over  the  simple  girder  for  long 
spans,  but  combines  with  it  the  kind  of  superiority  that  the  dome  has 
over  the  simple  arch  by  reason  of  circumferential  stresses  called  into 
play,  which  adds  greatly  to  the  carrying  capacity  of  the  slab. 

In  the  standard  mushroom  type,  which  is  quite  fully  discussed 
in  this  paper,  the  heavy  frame  work,  concentric  with  the  column, 
supports  the  slab  reinforcement'at  a  fixed  elevation,  furnishes  a  high 
degree  of  resistance  to  shear,  and  secures  a  high  degree  of  safety 
during  construction.  It  extends  as  a  cantilever  approximately  one 
fourth  of  the  way  to  the  next  column  as  shown  in  Figs.  1  and  2  on 
page  2.  Arranged  upon  the  radial  rods  of  this  frame  rest  two  or 
more  large  hoops  and  upon  these  rest  the  wide  spreading  belts  of  rods 
which  extend  both  directly  and  diagonally  from  column  to  column. 
Over  the  columns  these  belts  lie  near  the  upper  surface  of  the  slab,  but 
they  run  near  the  lower  surface  as  they  approach  points  midway 
between  columns. 

The  cantilever  slab  thus  formed,  not  only  has  the  same  advant- 
ages for  this  form  of  construction  that  the  cantilever  construction 
has  for  long  span  bridges,  but  it  causes  the  slab  to  have  greater 
stiffness  and  gives  it  greater  resistance  to  shear  in  the  neighborhood 
of  the  columns;  it  removes  the  locus  of  zero  bending  moment  to  a 
much  greater  distance  from  the  column  than  would  otherwise  be  the 
case,  thus  dimininishing  the  area  of  that  part  of  the  slab  which  tends 
to  become  concave  on  its  upper  face  and  enlarging  the  convex  area. 

The  cantilever  frame-work  further,  not  only  moves  the  locus  of 
zero  bending  outward  from  the  column,  but  it  also  fixes  the  locus  of 
zero  bending  rnoment  at  a  known  position  so  that  it  does  not  vary 
with  increase  and  decrease  of  the  load  or  change  of  the  load  from  one 
span  to  an  adjacent  span  as  would  be  the  case  were  the  mass  of 
metal  in  the  frame  and  its  stiffness  largely  reduced.  This  is  ac- 
complished as  follows: 

The  locus  of  zero  bending  moments  is  fixed  by  the  dip  of  the 
reinforceing  rods  as  they  leave  the  upper  surface  of  the  slab  near 
the  edge  of  mushroom  and  pass  below  the  neutral  surface  to  a  level 
near  the  bottom  of  the  slab.  Such  change  of  tensile  resistance  in 
the  slab  necessarily  localizes  at  these  points  the  zero  bending  mo- 
ments. 


4  SLAB    ACTION   BIBLIOGRAPHY 

In  addition  to  the  advantages  just  mentioned,  which  are 
of  so  self-evident  a  character  as  to  be  readily  appreciated  even 
by  the  layman,  there  is  another  of  such  an  obscure  and  apparently 
inexplicable  a  nature  that  it  was  for  years  denied  as  incredible  and 
regarded  as  non-existent  by  practical  builders,  and  engineers  as 
well,  unless  they  had  opportunity  to  be  convinced  of  its  reality 
by  experiment.  I  refer  here  to  the  additional  strength  and  stiff- 
ness which  is  imparted  to  a  belt  of  rods  in  a  given  direction  in  a 
slab  by  another  belt  at  right  angles  to  the  first  belt,  or  at  various 
angles  with  it.  This  should  be  designated  as  slab  action  proper 
in  distinction  from  cantilever  action.  It  depends  for  its  amount 
upon  the  value  of  Poisson's  ratio  of  lateral  contraction  to  direct 
elongation  in  the  slab,  and  is  the  basis  of  the  so  called  circum- 
ferential stresses,  which  make  the  strength  and  stiffness  of  such 
reinforced  flat  slabs  much  greater  than  they  are  estimated  to  be 
when  these  are  neglected,  as  they  usually  have  been.  This  mis- 
taken view  has  in  the  past  constituted  the  most  serious  obstacle 
to  the  adoption  of  this  form  of  structure,  and  has  been  the  ground 
of  conscientious  opposition  to  its  introduction  on  the  part  of  con- 
sulting engineers.  It  is  the  object  of  this  investigation  to  remove 
if  possible  all  reasonable  uncertainty  as  to  the  rational  theory  of 
this  form  of  structure. 

The  following  partial  bibliography  of  this  subject  may  be  useful 
to  those  unfamiliar  with  what  has  been  done  in  this  field. 

Concrete  Steel  Construction,  (305pp) 

By  C.  A.  P.  Turner,  M.  Am,  Soc.  C.  E. 
816  Phoenix  Bldg.,  Minneapolis,  1909. 

Reinforced  Concrete  Construction,  (259pp) 

By  Turneaure  and  Maurer,  University  of  Wisconsin 
Wiley,  N.  Y.,  1907. 

Concrete,  Plain  and  Reinforced,  (483pp) 
By  Taylor  and  Thompson, 
Wiley,  N.  Y.,  1911. 

Trans.  Am.  Soc.  C.  E. 
Vol.  LVI.  June  1906. 

Engineering  News: — 
Oct.  4,  1906,  p.  361. 
Feb.  18,  1909,  p  176. 
Dec.  23,  1909,  p.  694. 

Engineering  Record: — • 

March  28,  1908,  p  374. 
May  2,  1908,  p  575. 
Oct.  10,  1909,  p  411. 
April  3,  1909,  p  408. 
April  10  .1909,  p  492. 


NOTATION 

2.     All  lengths  and  areas  are  measured  in  inches,  and  all  weights 
in  pounds. 
A    =  area  of  cross  section  of  steel  reinforcement  per  unit  width  of 

slab,  in  case  it  be  assumed  to  be  replaced  by  a  uniform  sheet 

of  equal  weight. 

AI  =   area  of  cross  section  of  all  the  rods  in  one  side  belt. 
A2  =  area  of  cross  section  of  all  the  rods  in  one  diagonal  belt. 
a     =  one  half  the  longer  side  of  a  panel  from  center  to  center  of 

columns. 

b     =  one  half  the  shorter  side  of  a  panel. 
B    =  the  shortest  distance  along  one  side  of  a  panel  from  the  edge 

of  a  column  cap  to  the  edge  of  the  next  cap. 
Ci  and  C2  are  constants  depending  on  the  relative  lengths  of  the  sides 

of  any  panel,  which  reduce  to  unity  for  any  square  panel. 
DI  =  the  deflection  of  the  middle  of  the  longer  side  of  the  panel 

below  the  edge  of  the  cap. 
D2  =  the  deflection  of  the  center  of  the  panel  below  the  edge  of  the 

cap. 
d     =  the  effective  thickness  of  the  slab  at  any  point,  being  the 

vertical  distance  from  the  center  of  action  of  the  reinforce- 
ment to  the  compressed  surface  of  the  concrete. 
di    =  the  vertical  distance  from  the  center  of  the  rods  in  the  side 

belt  at  mid  span  to  the  top  surface  of  the  concrete. 
d2    =  the  distance  at  the  center  of  the  panel  from  the  center  of  the 

rods  in  the  second  or  upper  diagonal  belt  to  the    top    of    the 

concrete. 
d3    =  the  distance  at  the  edge  of  the  cap  from  the  center  of  the  third 

belt  of  rods  from  the  top,  to  the  compressed  surface  of  the 

concrete. 

E  or  Ea  =   Young's  modulus  for  steel   =   3  x  107. 
Ec  =  Young's  modulus  for  concrete. 
ei    =  elongation  in  steel  parallel  to  long  side  belt. 
e2    =  elongation  in  steel  parallel  to  short  side  belt. 
61     =  elongation  in  steel  parallel  to  diagonal  belt. 


NOTATION 


F    =  modulus  of  elastic  resistance  to  shearing. 

js     =  Ee  =  intensity  of  actual  stress  in  steel. 

/c    =  intensity  of  stress  in  concrete. 

g     =   7/16  (a+6)  =  the  diameter  of  the  mushroom  head  and  width 

of  belts. 

h     =  the  total  actual  thickness  of  concrete  slab. 
id    =  vertical  distance  from  center  of  tension  of  steel  to  neutral 

surface  of  slab. 
jd    =  vertical  distance  from  center  of  tension  in  steel  to  center  of 

compression  in  concrete. 
Jed  =  vertical  distance  from  neutral  surface  to  compressed  surface 

of  concrete,  hence  i  +  k  =  1. 

K    =  Poisson's  ratio  of  lateral  contraction  to  longitudinal  stretch- 
ing for  reinforced  concrete  slabs. 

LI   =  2a  =  long  side  of  panel  between  column  centers. 
L2   =   26   =  short  side  of  panel  between  column  centers. 
I     =   distance  from  collar  band  at  top  of  column  to  edge  of  .cap. 
mi  =  true  moment  of  resistance  of  the  tensile  stresses  in  steel  parallel 

to  the  long  side  per  unit  of  width  of  slab. 
m<2  =  true  moment  of  resistance  of  steel  parallel  to  short  side  per 

unit  of  width. 

nil  and  m2  =   apparent  moments  per  unit  of  width  of  forces  applied 

parallel  to  the  long  and  short  sides  respectively, 
n  =  the  apparent  moments  per  unit  of  width  of  the  equal 

twisting  couples  parallel  to  either  side. 

Pi  =  intensity  of  the  forces  applied  parallel  to  the  long  side. 

p2  =   ditto  for  short  side. 

p  =  intensity  of  stress  in  extreme  fiber  of  radial  rods. 

q  =  load  on  slab  in  pounds  per  square  inch. 

RI  and  R2    =  the  radii  of  curvature  of  vertical  sections  of  the  slab 

parallel  to  the  long  and  short  sides  respectively. 
Si  and  s2      =  the  vertical  shearing  stresses  per  unit  of  width  of  slab 

respectively  perpendicular  to  the  long  and  short  sides 

of  the  slab. 
<s  =  the  intensity  of  vertical    shearing    stress    in    radial 

rods, 
t  =  either  of  the  equal  horizontal  tangential  or  shearing 

stresses  parallel  to  the  sides  of  the  panel. 


NOTATION.       SQUARE    PANEL 


t 

u  and  v 

V 

x  y  z 


Az 

zi  and  z2 

8 

5z 

5x 


=  the  thickness  of  a  radial  rod. 

=   deformations  parallel  to  the  long  and  short  sides  re- 
spectively. 

=  total  vertical  shearing  stress  in  radial  rod. 

=  horizontal  and  vertical  coordinates  parallel  to  sides 

of  panel. 

=   difference  of  two  vertical  coordinates. 
=   deflections  of  radial  rods. 
=  sign  of  partial  differential. 

=  partial  differential  coefficient  of  z  with  respect  to  x. 


'/<//** 

/  xN/X 


X    X  x  X   X   X   X 


XX 


ir\^ 


>\*S/ 


7*  /  / 


X 

X 

x 
\x 


s 


\x 

^ 


\ 

\/r 
x> 

X}. 


x/ 


XiX  WiS 


s>< 


^ 


XP 


a* 


ft  2! 


St 


7^ 


25  X 


^ 


\yx- 


Fig.  3.     Plan  of  Reinforcement  Mushroom  System. 
Square  Panel,  g  =  %L  (as  drawn). 
Line  of  Ultimate  Weakness. 


8  TRUE    AND    APPARENT   STRESSES 

3.  As  preliminary  to  a  general  investigation  of  the  rational 
analysis  of  the  flat  slab,  it  seems  desirable  in  the  first  place  to 
make  a  brief  exposition  of  the  relationship  between  the  true  bend- 
ing moments  and  the  apparent  bending  moments  in  the  flat  slab  as 
follows: 

The  fundamental  equations  of  extensional  stress  and  strain  in 
thin  flat  plates  and  slabs,  established  a  generation  ago  and  accepted 
by  Grashof*and  by  all  authorities  on  the  subject  since  then,  maybe 
written  in  the  forms  : 


Eei  =  Pi 

Ee2  =  p2  —  Kpi  ) 

(l-K2)Pl=E(e1+Ke2)\ 
(1-K2)P2  =  E(e2  +  Kel}  ^ 

in  which  pi  and  p2  are  the  external  applied  or  apparent  stresses  per 
unit  of  area  of  cross  section  of  the  plate,  or  of  the  reinforced  slab, 
which  act  parallel  to  the  axes  of  x  and  y  respectively  if  these  latter 
lie  in  the  neutral  plane  of  the  slab;  and  BI  and  e2  are  extensometer 
elongations  of  plate  or  slab  reinforcement  per  unit  of  length  parallel 
to  x  and  y  respectively.  E  is  Young's  modulus  of  elasticity,  and  K 
is  Poisson's  ratio  of  lateral  contraction  to  linear  elongation.  Any 
piece  of  material  which  is  subjected  to  stress,  and  is  of  such  shape 
that  more  than  one  of  its  dimensions  is  considerable,  as  compared 
with  its  remaining  dimension,  must  have  its  stresses  and  strains 
considered  with  reference  to  lateral  contraction.  This  is  the  case 
in  plates  and  slabs,  as  it  is  not  in  case  of  rods  and  beams. 

In  the  above  equations  Eei  and  Ee2  are  the  true  stresses  per  square 
inch  of  section  of  reinforcement  acting  along  lines  parallel  to  x  and  y 
respectively,  whatever  pi  and  p2  may  be.  These  latter  are  the  cause 
of  true  stresses,  but  are  not  themselves  the  values  of  the  true  stresses, 
as  they  are  in  case  of  rods,  etc.,  where  one  dimension  only  is  large. 

These  equations  show  that  the  elongation  e^  in  the  direction  of 
x  is  not  dependent  alone  upon  the  tension  pi  applied  in  that  direc- 
tion, for  it  is  diminished  by  any  tension  acting  along  y,  but  is  in- 
creased by  any  compression  acting  along  y.  It  thus  appears  that 
any  tension  p2  along  y  assists  the  piece  in  resisting  elongation  along 
x  and  makes  it  able  to  endure  safely  a  larger  applied  stress  pi  with 
the  same  degree  of  safety,  i.  e.,  with  the  same  percentages  of  elonga- 
tion or  true  stress.  But  it  is  also  equally  true  that  any  compression 
of  amount  p2  reduces  the  safe  value  of  pi  which  may  be  applied  to 

*Theorie  der  Elasticitat  und  Festgkeit,  F.  Grashof  Berlin  1878. 


TRUE    AND    APPARENT   MOMENTS  9 

it.  These  principles  are  not  in  accordance  with  those  which  hold 
in  ordinary  computations  for  rods  and  bars,  whose  lateral  dimensions 
are  small'  compared  with  their  lengths,  and  whose  lateral  stresses 
are  negligible.  This  divergence  between  the  true  stresses  as  shown 
by  actual  deformations,  and  the  apparent  or  applied  stresses,  is  a 
fruitful  source  of  error  in  the  attempted  computation  of  slabs. 

Equations  (1)  in  their  present  form  apply  to  simple  extensional 
or  compressive  stresses  and  strains  but  may  be  extended  to  apply 
to  bending  of  slabs  in  the  following  manner: 

Take  A  as  the  cross  section  of  the  reinforcement  per  unit  of 
width  of  slab  when  the  actual  reinforcement  is  regarded  as  distrib- 
uted into  a  thin  sheet  of  uniform  thickness,  and  let  jd  be  the  vertical 
distance  from  the  center  of  the  reinforcement  to  the  center  of  com- 
pressional  resistance  of  the  concrete  regarded  as  a  fraction  j  of  d, 
d  being  the  distance  from  the  center  of  the  steel  to  the  top  of  the 
slab.  Then 

mi  =  Api  jd,  and  m2  =  Ap2  jd,  ...................  (2) 

are  the  apparent  bending  moments  per  unit  of  width  of  slab,  of  the 
applied  apparent  stresses  pi  and  p2,  tending  when  positive,  to  cause 
lines  which  before  bending  are  straight  and  parallel  to  x  and  y  re- 
spectively, to  become  concave  upwards. 

Again  mi  =  Ee\  Ajd,  and  m2  =  Ee2  Aid,  .................  (3) 

are  the  true  bending  moments  of  the  actual  resistance  stresses  in 
the  reinforcement  per  unit  of  width  of  slab,  as  shown  by  extenso- 
meter  strains  in  the  steel  parallel  to  the  axes  of  x  and  y  respectively. 
Multiply  equations  (1)  thru  by  Ajd  and  substitute  the  values 
given  in  equations  (2)  and  (3),  from  which  we  obtain  the  following 
relations  between  the  true  and  apparent  bending  moments  in  the  slab. 

mi  =  mi  —  Km2\  ^  ^ 

m2  =  m2  —  Kml^ 

(1  —  K2)mi  =  mi  +  Km2  \  (^ 

(l—K2)m2  =  m2 


These  equations  bring  out  in  a  striking  manner  the  essential  diver- 
gence of  the  correct  theory  of  slab  action  from  that  of  beam  action 
in  which  latter  case  we  have  the  well  known  equations 

mi  =  m>ij  and  m2  =  m2 

i.  e.,  in  beams  the  moment  of  the  applied  forces  is  equal  to  the 
moment  of  the  internal  resistance,  which  is  not  true  in  slabs. 


10  TRUE    AND    APPARENT   MOMENTS 

All  attempts  to  base  computations  of  the  deflection  of  slabs 
upon  beam  action  are  therefore  necessarily  erroneous.  Such  com- 
putations are  inapplicable  and  misleading,  hence  deflections  and 
stresses  in  slabs  cannot  be  correctly  computed  by  any  form  of 
simple  or  compound  beam  theory. 

Equations  (4)  show: 

1st  That  at  points  where  n^  and  m2  are  of  the  same  sign,  (as 
for  example  in  the  convex  part  of  the  mushroom  near  the  columns 
and  also  near  the  center  of  the  panel)  the  true  bending  moments 
mi  and  m2,  which  determine  the  actual  stresses  in  the  reinforcement 
are  less  than  the  apparent  bending  moments,  which  latter  have  been 
ordinarily  assumed,  according  to  the  beam  theory,  to  determine 
those  stresses. 

2nd  That  the  compressive  stresses  in  the  concrete  around  the 
column  cap  are  determined  on  the  same  principles  as  the  tensile 
stresses  and  are  consequently  reduced  in  accordance  with  the  value 
of  K  by  a  considerable  percentage  below  values  corresponding  to 
ni!  and  m2  of  the  beam  theory. 

3rd  That  at  points  where  n^  and  m2  have  different  signs,  as 
they  have  for  example  in  the  middle  part  of  the  span  at  the  side  of 
the  panel  directly  between  mushroom  heads,  the  values  of  the  true 
bending  moments  are  larger  than  the  apparent  moments  as  found 
by  the  beam  theory. 

4th  One  deduction  from  this  (which  is  also  confirmed  by 
extensometer  tests)  is,  that  in  slabs  having  equal  side  and  diagonal 
belts  of  reinforcing  rods  the  greatest  actual  extensions  and  true 
stresses  in  the  steel  occur  at  the  mid  points  of  those  reinforcing  rods 
which  run  directly  between  the  mushroom  heads  parallel  to  the 
sides  of  the  panel,  and  do  not  occur  at  the  center  of  the  panel  where 
nix  and  m2  have  their  greatest  values.  Further,  the  true  stresses  in 
the  reinforcement  are  not  so  large  at  the  edge  of  the  column  caps  as 
at  the  points  just  indicated.  Neither  of  these  conclusions  is  in 
accordance  with  the  beam  theory  as  implied  in  ordinary  formulas 
such  as  have  been  frequently  adopted  in  practice  in  computing  slabs. 

5th  In  making  any  statement  or  specification  respecting  the 
bending  moments  at  any  point  of  a  slab,  it  is  essential  to  state  which 
bending  moments  are  contemplated,  the  true  bending  moments  or 
the  apparent  moments,  with  the  understanding  that  the  true  bend- 
ing moments  only  are  to  be  used  in  determining  cross  sections  and 
stresses  of  steel.  Any  statement  omitting  this  distinction  is  ambig- 
uous, and  any  requirement  seeking  to  proportion  cross  sections  of 
steel  to  apparent  stresses  and  apparent  moments  is  incorrect. 


POISSON'S  RATIO  11 

4.  Poisson's  ratio  K  plays  an  important  role  in  the  theory  of 
flat  slabs  and  plates,  as  is  evident  from  equations  (1)  and  (4).  Few 
attempts  have  been  made  to  determine  K  by  directly  measuring  the 
amount  of  the  lateral  contraction  accompanying  the  elongation  of 
test  specimens,  and,  were  such  measurements  made,  the  relative 
dimensions  of  the  cross  section  of  the  specimen  would  need  to  be 
considered  as  affecting  in  a  very  complicated  way  the  true  value  of 
K  to  be  derived  from  observation.  Reliable  determinations  of  K 
usually  depend  upon  observations  of  Young's  modulus  of  elasticity 
E  and  the  shearing  modulus  of  elasticity  F. 

It  is  proven  in  the  general  theory  of  the  deformation  of  isotropic 
elastic  solids  that  all  the  elastic  properties  of  any  such  solid  are 
determined  without  excess  or  defect  by  its  values  of  E  and  Ft  and 
that  Poisson's  ratio  is  a  function  of  E  and  F  expressed  by  the  equation 

K  +  l  =  \E/F (5) 

There  is  evidence  to  show  that  for  concrete  K  is  approximately 
0.1*.  For  steel  it  is  known  that  K  =  0.3  nearly. 

Now  it  is  evident  that  a  horizontal  slab  of  reinforced  concrete, 
in  which  the  reinforcement  consists  of  rods,  differs  from  one  in  which 
its  reinforcement  is  considered  to  be  a  simple  uniform  sheet  of  metal 
in  this,  that  the  former  has  much  less  shearing  rigidity  in  resisting 
horizontal  forces  than  the  latter,  for  in  it  all  stresses  transmitted 
from  one  band  or  belt  of  rods  to  any  other  belt  crossing  it  are  trans- 
mitted thru  concrete  only,  as  is  not  the  case  if  the  reinforcement 
consists  of  a  continuous  sheet.  It  is  evident  therefore  that  the  value 
of  K  which  must  be  employed  in  applying  the  foregoing  equations  to 
reinforced  concrete  slabs  must  exceed  0.3,  the  value  required  in  case 
the  reinforcement  is  a  sheet  of  steel. 

This  analysis  of  the  conditions  affecting  the  value  of  K  for  a 
reinforced  flat  slab  differs  radically  from  assuming  at  ramdon  that 
because  K  =  0.3  for  steel  alone  and  K  =  0.1  possibly,  for  concrete 
alone,  that  therefore  some  intermediate  value  of  K  may  be  correct 
for  these  two  materials  combined  in  a  slab.  Such  an  assumption 
is  merely  a  blind  guess  and  has  no  rational  basis. 

As  already  partly  stated,  the  view  here  put  forth  is  this :  Since 
in  any  homogeneous,  isotropic,  elastic  material  the  experimental 
values  of  E  and  F  perfectly  define  all  its  elastic  properties,  and  since 
we  are  evidently  at  liberty  to  assume  our  flat  slab  as  sufficiently  fine 
grained  in  its  structure  to  act  nearly  like  a  slab  constructed  of  some 
sort  of  homogeneous  materials,  it  will  be  possible  to  determine 

*  Turneaure  and  Maurer's  Reinforced  Concrete  Construction  2nd  Ed.  1907,  p.  210. 


12  -POISSON'S  RATIO 

certain  mean  values  of  E  and  F  which  will  define  its  elastic  proper- 
ties. It  is  moreover  evident  that  in  a  slab,  where  two  kinds  of  elastic 
solids  are  combined  as  they  are  here,  the  mean  value  of  F  for  the 
combination  is  much  more  affected  by  the  concrete  than  is  E,  which 
latter  may  be  taken  as  that  applying  to  the  steel  alone,  and,  conse- 
quently as  unchanged  by  the  combination.  It  is  otherwise,  however, 
with  F,  because  the  arrangement  of  the  combination  is  such  as  to 
require  the  assumption  of  a  value  of  F  lying  somewhere  between 
that  for  steel  and  that  for  concrete.  Since  the  latter  value  is  much 
less  than  the  former,  the  mean  value  of  F  is  smaller  than  for  steel 
alone. 

This  reasoning  and  other  independent  theoretical  and  kinemat- 
ical  considerations  have  led  to  the  same  conclusion,  viz:  that  the 
correct  value  of  K  for  the  slab  is  larger  than  0.3. 

Assuming  E  =  30,000,000,  we  may  compute  corresponding 
values  of  K  and  F  from  (5)  as  follows: — 

If  K  =  0.1  ,  F  =  13,600,000 
If  #  =  0.3  ,  F  =  11,600,000 
If  K  =  0.5  ,  F  =  10,000,000 

Were  a  perfectly  complete  and  accurate  mathematical  theory 
of  the  flat  slab  at  our  disposal,  we  might  consider  every  experimental 
test  of  the  deflection  of  such  a  slab,  and  every  extensometer  measure- 
ment of  its  reinforcing  rods  as  an  experiment  for  determining  the 
numerical  value  of  K,  since  deflections  and  extensions  would  then 
all  be  known  functions  of  K.  Having  brought  such  a  rational 
theory  to  a  somewhat  satisfactory  degree  of  perfection,  the  writer 
finds  that,  in  the  light  of  all  known  tests  of  slabs,  the  value  that  best 

satisfies  all  conditions  is      K  =  0.5 (6) 

It  is  possible  that  this  value  of  the  constant  K  for  slabs  may  need 
some  slight  modifications  hereafter,  but  for  the  present  this  may  be 
regarded  as  substantially  correct  for  mushroom  slabs.  It  may  be 
found  necessary  to  assume  a  somewhat  different  value  for  other  forms 
of  structure,  as  for  example,  beam  and  girder  construction.  That, 
however,  must  be  determined  later.  Moreover  it  must  be  said  that 
this  value  of  K  applies  to  tests  made  upon  slabs  from  2  to  4  months 
old,  and  under  loads  which  have  been  applied  to  such  relatively  soft 
concrete  as  this  for  a  period  of  usually  not  longer  than  one  or  two 
days,  and  of  an  intensity  such  as  to  cause  a  maximum  stress  in  the 
steel  of  from  10,000  to  16,000  Ibs.  per  square  inch.  Less  loads  on 
better  cured  concrete,  or  longer  time  under  load,  may  show  con- 
siderable deviation  from  this  value  of  K. 


EQUILIBRIUM    OF    SLAB    ELEMENT  13 

How  important  a  factor  K  is  in  slab  theory  is  evident  on  con- 
sidering equations  (4)  which  show  that  in  a  square  panel  uniformly 
loaded  the  true  moments  as  shown  by  the  elongations  of  the  rein- 
forcing rods  at  the  center  of  the  panel  and  over  the  centers  of  the 
columns  are  only  one  half  the  corresponding  apparent  moments 
derived  from  considering  the  moments  required  to  hold  the  applied 
forces  in  equilibrium,  this  being  on  the  assumption  of  course  that 
K  =  0.5. 

5.  In  order  to  derive  the  general  differential  equation  of  shears 
and  moments  in  any  rectangular  panel  in  an  extended  horizontal 
plate  or  slab,  take  the  axes  of  x  and  y  in  the  neutral  plane  of  the 
plate  and  parallel  respectively  to  the  longer  and  shorter  sides  of  the 
panel  with  the  origin  at  its  center  before  flexure  occurs,  and  assume 
that  they  remain  fixed  with  reference  to  the  points  of  support  of  the 
panel.  Then  during  flexure  the  center  of  the  panel  and  all  other 
points  of  the  slab  or  plate  not  in  contact  with  the  fixed  points  of 
support  will  attain  some  deflection  2,  of  amount  to  be  determined 
later.  Take  z  positive  downwards. 

Then  dxdy  is  the  horizontal  area  of  an  element  of  the  slab 
bounded  by  vertical  planes,  and  if  d  be  the  effective  thickness  of  the 
slab  or  plate,  the  areas  of  the  sides  of  this  element  which  are  respec- 
tively perpendicular  to  x  and  y  are  ddy  and  ddx,  while  d8xdy  is  the 
volume  of  the  element. 

We  proceed  to  obtain  the  equations  of  equilibrium  of  this  ele- 
ment of  the  slab  as  follows : — 

Let  Si  and  s2  be  the  total  vertical  shearing  stresses  per  unit  of 
width  of  slab  for  sections  perpendicular  to  x  and  y  respectively.  In 
case  these  shears  are  variable,  as  they  are  in  a  continuously  loaded 
slab,  they  respectively  contribute  elementary  forces  tending  to  move 
the  element  vertically,  of  the  following  amounts : 

d  &  d  s2 

dydx,         and  -  dxdy 


d  x  8  y 

Assume  that  the  slab  carries  a  uniformly  distributed  load  of  q  pounds 
per  square  unit  of  area.  Then  the  load  upon  the  elementary  area 
dxdy  is  qdxdy}  and  the  equation  of  equilibrium  of  the  vertical  forces 
acting  on  the  element  reduces  to  this: 

d  Si  d  s2 

+  +<?  =  0 (7) 

8  x  ft  y 


14  EQUILIBRIUM    OF    SLAB    ELEMENT 

in  which  Si  and  s2  are  taken  as  positive  when  they  are  such  as  would 
be  produced  in  the  slab  by  the  loading  q  in  case  it  were  supported  at 
the  origin  only. 

Let  nii  and  m2  be  the  apparent  moments  per  unit  of  width  of 
slab  of  the  applied  forces  which  tend  to  bend  those  lines  in  the  slab 
which  before  bending  are  parallel  to  x  and  y  respectively.  Take 
them  as  positive  when  they  tend  to  make  those  lines  respectively 
concave  upwards.  These  are  the  moments  obtained  by  multiplying 
the  total  applied  tension  per  unit  of  width  of  slab  by  the  vertical 
distance  jd  from  the  center  of  the  reinforcement  of  the  slab  to  the 
center  of  compression  in  the  concrete  as  given  in  (2) .  These  moments 
are  not  identical  in  a  slab  with  the  true  resisting  moments  mi  and  w2 
in  the  same  directions,  which  latter  are  the  moments  obtained  by 
multiplying  jd  by  the  actual  tension  in  the  steel  per  unit  of  width  of 
slab,  which  last  is  to  be  correctly  computed  by  taking  the  product 
of  the  area  of  steel  per  unit  of  width  and  its  elongation  multiplied 
by  E  its  modulus  of  elasticity  as  shown  in  (3). 

Again,  let  n  be  the  twisting  moment  per  unit  of  width  of  ver- 
tical section  of  slab  cut  by  planes  perpendicular  to  either  x  or  y,  and 
acting  about  either  x  or  y,  which  moment  n  is  regarded  as  due  to  the 
variation  of  the  vertical  shearing  stress  Si  when  y  varies,  and  to  the 
variation  of  S2  when  x  varies.  The  moment  n  is  held  in  equilibrium 
by  horizontal  shearing  stresses  in  these  same  sections,  which  are 
opposite  in  sign  above  and  below  the  neutral  surface.  Let  t  be  the 
total  horizontal  shearing  stress  per  unit  of  width  of  slab  in  the  rein- 
forcement on  one  side  of  the  neutral  plane,  then: 

n  =  t  A  j  d (8) 

At  any  point  xy  this  horizontal  shearing  stress  t  must  be  the  same 
fpr  the  section  perpendicular  to  x,  as  for  the  section  perpendicular 
to  y,  because  in  every  state  of  stress  the  tangential  components  are 
equal  and  of  opposite  sign  on  any  two  planes  mutually  at  right 
angles.  Consequently  the  moment  n  is  the  same  about  x  as  about 
y,  as  has  been  assumed  in  (8). 

It  is  implicitly  assumed  in  (2)  and  (3)  that  the  concrete  on  the 
same  side  of  the  neutral  plane  as  the  reinforcement  is  ineffective 
and  that  its  resistance  is  negligible,  so  that  on  that  side  the  resistance 
of  the  reinforcement  alone  counts.  This  condition  actually  occurs 
only  after  a  state  of  quite  considerable  stress  obtains,  and  of  itself 
affords  a  sufficient  reason  why  the  formulas  based  on  it  fail  of  accu- 
rately representing  deflections  and  elongations  at  small  loads  and 
low  stress. 


DIFFERENTIAL   EQUATION    OF    MOMENTS  15 

The  elementary  couples  acting  on  the  vertical  faces  of  the 
element  which  are  in  equilibrium  with  those  arising  from  the  shear- 
ing stresses  are: — 

(5  m1  5  n    \ 

-  -J-  -          -  I     dxdy     about  y,  and 
d  x  by/ 

(b  m2  d  n    \ 
-j-  -  I     dxdy     about  x; 
d  y              d  x    / 

while  those  arising  from  the  shears  themselves  are: — 
Si  dx  by     and     s2  bx  by. 

Consequently  the  equations  of  equilibrium  of  the  couples  acting  on 
the  element  reduce  to  the  following: 


(9) 


Differentiate  equations  (9)  with  respect  to  x  and  y  respectively 
and  substitute  in  (7),  and  we  obtain 

<52mi  52  n  62m2 

+  -  =   q (10) 


d 

nil 

b 

n 

e                O 

d 

x 

_ 

b 

y 

. 

Si    - 

d 

m2 

b 

n 

a                 fl 

d 

y 

b 

x 

S2    —    U 

d  x2  dx  by  b  y2 

which  is  a  general  differential  equation  of  the  apparent  moments 
of  the  applied  forces  which  exist  in  a  uniformly  loaded  slab  in 
terms  of  rectangular  coordinates.  From  it  the  differential  equa- 
tion of  the  deflections  may  be  derived  as  follows: — 

6.  To  obtain  the  general  differential  equation  of  the  deflec- 
tions of  a  slab,  note  that  from  geometrical  considerations  such  as 
are  familiar  in  the  theory  of  beams  we  have 

RI€I  =  id  =  R2e2 (11) 

in  which  RI  and  R2  are  the  radii  of  curvature  of  sections  of  the 
neutral  surface  by  vertical  planes  parallel  to  x  and  y  respectively; 
and  id  is  the  distance  from  the  center  of  the  reinforcement  to  the 


16 


MOMENTS    AND    CURVATURES    IN    SLAB 


neutral  surface.     In  equations   (la)  replace  pi  and  p2  by  values 
given  in  (2),  and  ei  and  e2  by  values  taken  from  (11)  and  we  have: — 


(1  —  irtai  =  E  A  ijd2  I  — 


(12) 
(1  —  #2)m2  =  E  A 

But  from  the  theory  of  curvature 

1  52  z  1  d2  z 

—  =  Z  -       -,  and  -  -  =  Z 

#1  5  Z2  #2  5  / 

Also  write  for  brevity        I  =  A  i  j  d2 (14) 

Then  we  have  from  (12),  (13)  and  (14): 

=  ±_EI(^-Z~  +K- 
5  x2  d 

....  (15) 

d  yr 

By  the  fundamental  equations  of  elasticity  we  also  have 

(du       5v\ 
—  +  —  1 (16) 

In  which  F  is  the  shearing  modulus,  e3  is  the  horizontal  shearing 
deformation  of  the  reinforcement  for  two  vertical  planes  one  unit 
apart  horizontally,  and 

i         5z  i          dz 

u  =  ZLid—    ,    v  =  ±id— (17) 

dx  by 

are  the  deformations  along  x  and  y  respectively,  due  to  the  vertical 
•distance  i  d  of  the  reinforcement  from  the  neutral  surface. 

From  (16)  by  help  of  (17)  we  have 

d2  z 

t  =  ±2Fid-        - (18) 

5x8y 


DIFFERENTIAL   EQUATION   OF    DEFLECTIONS  17 

In  (18)  replace  F  by  its  value  obtained  from  (5),  and  then  sub- 
stitute the  resulting  value  of  t  in  (8) : — 
we  then  have 

E  I         52  z 

n  =  -         -•-       - (19) 

1  +  K      dxdy 

From  (15)  and  (19)  obtain  values  of  the  second  differential 
coefficients  of  the  moments  appearing  in  (10),  which  on  being  intro- 
duced into  (10),  transform  that  equation  into  the  required  general 
differential  equation  of  deflections  as  follows: — 

54  z  d4  z  54  z          (1  —  K2) 

T^  +  W+7^  =  ^^ (20) 

which  is  a  partial  differential  equation  of  the  fourth  order  that  must 
be  satisfied  by  the  coordinates  x  y  z  of  the  neutral  surface  of  any 
uniform  plate  or  slab  initially  flat,  when  deflected  by  the  applica- 
tion of  a  uniformly  distributed  load  of  intensity  g,  and  supported  in 
any  manner  whatever. 

It  may  be  shown  that  any  deviations  from  strict  accuracy  by 
reason  of  local  stretching  of  the  neutral  surface  (here  neglected)  are 
small  compared  with  corresponding  deviations  in  beam  theory. 

7.  The  solution  of  the  general  differential  equation  of  deflec- 
tions (20)  for  the  case  of  a  horizontal  slab  carrying  a  uniformly 
distributed  load  and  supported  on  rows  of  columns  placed  in  rec- 
tangular array  and  having  the  points  of  support  all  on  the  same 
level,  will  now  be  considered. 

The  integration  or  solution  of  (20)  would,  since  it  is  a  partial 
differential  equation,  introduce  arbitrary  functions  of  the  independent 
variables  x  and  y  whose  forms  would  need  to  be  so  determined  as  to 
cause  the  solution  to  satisfy  the  conditions  imposed  by  the  position 
and  character  of  the  supports  at  certain  points,  or  along  certain 
lines.  It  would  be  possible  to  expand  these  functions  in  terms  of 
ascending  whole  powers  and  products  of  x  and  y,  and,  in  case  the 
supports  are  symmetrically  situated  with  respect  to  the  axes,  the 
expansions  will  contain  no  odd  powers  of  x  or  y,  because  the  value 
of  z  must  remain  unchanged  by  changes  of  sign  of  either  x  or  y,  or 
both  x  and  y.  Any  form  of  polynomial  expansion  which  satisfies 
(20),  and  also  all  the  conditions  of  any  given  case,  must  be  the  correct 
solution  for  that  case,  for,  the  solution  of  any  given  case  must  be 
unique. 


18  GENERAL   EQUATION    OF   DEFLECTIONS 

Instead  therefore  of  carrying  thru  the  tedious  analytical  devel- 
opment involved  in  solving  (20)  mathematically  and  then  applying 
it  to  the  case  we  are  treating,  we  shall  at  once  write  down  the  form 
of  solution  that  applies  to  the  case  in  hand  and  verify  the  fact  that 
it  satisfies  (2)  and  all  the  required  geometrical  conditions.  It  will 
therefore  be  the  solution  sought  for,  which  might  also  have  been 
obtained  by  the  somewhat  intricate  analytical  processes  involved 
in  the  intregation  of  such  differential  equations  as  (20). 

Assuming  at  first  that  the  slab  is  unlimited  in  extent  and  uni- 
form thruout  in  the  distribution  of  its  reinforcement  and  loading, 
and  that  the  parallel  rows  of  supporting  columns  divide  the  slab 
into  equal  rectangular  panels,  we  shall  find  a  solution  in  which  every 
panel  is  deformed  precisely  in  the  same  manner  as  every  other. 
Modifications  made  later  will  render  it  possible  to  take  account  of 
variations  and  irregularities  in  the  distribution  and  arrangement  of 
the  reinforcement,  and  to  estimate  to  some  extent  at  least  the  effect 
of  loading  only  one  or  more  panels. 

Let  2a  be  the  length  and  2b  be  the  breadth  of  a  panel;  then  the 
equation  of  its  neutral  surface,  referred  to  axes  parallel  to  its  sides 
and  to  an  origin  fixed  in  space  at  the  center  of  the  neutral  surface  of 
the  panel  before  deflection,  is:— 

48  EIz  =  9(1  -  K2)  [(a2  -  x2)2  +  (b2  -y2)2]  .....  (21) 

This  is  the  correct  solution  of  (20)  not  only  because  it  satisfies 
(20),  as  it  will  be  found  to  do  by  trial,  (and  just  as  many  other  func- 
tions of  x  and  y  do  also)  but  it  also  satisfies  all  the  other  conditions 
required  by  the  case  proposed,  viz.  : 

1st    z  =  0    when  both  x  =  +.  a  and  y  =  +  6; 
because  there  must  be  no  deflection  at  these  points  of  support  which 
are  on  the  same  level  as  the  origin. 

2nd  dz  /  dx  =  0,  when  x  =  0,  and  also  when  x  =  i  a;  as  well  as 
dz/dy  =  Q,  when  y  =  0,  and  also  when  y  =  +_  b;  because  straight 
lines  drawn  in  space  to  touch  the  slab  across  its  edges,  and  across 
its  mid  sections  parallel  to  those  edges,  must  all  be  horizontal  by 
reason  of  the  symmetry  of  the  slab  on  each  side  of  its  edges  and  mid 
sections.  That  these  conditions  hold  is  evident  from  the  following 
equations  derived  from  (20)  : 


12  E  I 


(2  2, 

q  x  (x   --  a) 


-       -  qy 
12  El 


(22) 


TRUE    STRESSES    AND    MOMENTS.       LINES    OF    CONTRA-FLEXURE 


19 


It  is  of  interest  to  note  that  the  sections  of  this  surface  made 
by  all  vertical  planes  parallel  to  the  axes  of  y,  i.  e.,  by  x  =  constant, 
are  precisely  the  same  except  in  position,  since  their  equations  differ 
by  a  constant  only.  The  same  is  true  of  sections  parallel  to  x.  It 
thus  appears,  that,  in  a  square  panel  where  a  =  b,  the  surface  may 
be  regarded  as  a  ruled  surface  described  by  using  the  two  of  these 
curves  on  a  pair  of  parallel  sides  of  the  panel  as  directrices  and  a 
third  one  of  these  curves  as  a  ruler  sliding  on  the  first  two  in  such  a 
manner  as  to  remain  parallel  to  the  other  pair  of  parallel  sides. 

The  deflections  at  the  center  of  the  panel  and  middles  of  the 

sides  are: 

At  x  =  0  =  y,  48  E  I  z  =  q  (l-K2)  (a4  +  64) 

At  x  =  ±  a,  y  =  0,         48  E  I  z  =  q  (l-K2)  64 
At  x  =  0,  y  =  ±  b,         48  E  I  z  =  q  (l-K2)  a4 

so  that  in  a  square  panel  the  center  deflection  is  twice  the  mid  edge 

deflection. 

Differentiating  equations  (22)  we  have  by  help  of  (11),  (13), 
(14)  and  (3): 


id  52  z 

el  =  ~-  =  +  id- 
Ri  5x2 

52  z 


id 
€2  =  —  =  ±  id 


5x2 


12  E  A  j  d 

(l-K2) 
12  E  A  j  d 


....(23) 


q  (3y2  -  b2) 


12 


12 


q  (3x2  —  a2) 


q  (3y2  —  b2) 


(23a) 


in  which  the  ambiguous  signs  are  to  be  so  taken  that  mi  and  m2  in 
(15)  will  be  positive  at  x  =  0  =  y,  and  negative  at  x  =  +.  a  and 
y  =  ±b. 

From  (23)  it  appears  that  extensions  vanish  and  contra-flexure 
occurs  at  lines  lying  in  vertical  planes  whose  equations  are 

(24) 


It  thus  appears  that  the  slab  is  subdivided  by  these  lines  (24) 
drawn  parallel  to  the  edges  into  a  pattern  which  consists  of  a  rect- 
angle occupying  the  middle  part  of  each  panel,  of  a  size  f  aVs  by 


20  APPARENT   MOMENTS   AND    SHEARS 

f  bV%,  i.  e.,  of  the  same  relative  dimensions  as  the  panel  itself,  and 

bounded  by  lines  (24),  which  rectangle  is  concave  upward  thruout. 

On  all  four  sides  of  this  central  rectangle  are  rectangles  of  saddle 

shaped  curvature  directly  between  the  central  rectangles  of  adjoin- 

ing panels,  while  each  point  of  support  is  situated  in  a  rectangle 

which  is  convex  upward  over  its  entire  area,  of  dimensions 

2a(l—  J  ^3)  by  26(1  —  J 


From  (22)  we  obtain  the  equation 

32  z  /  dxdy  =  0  ...............................  (25), 

hence  by  (18)  and  (19)  it  follows  that 

t  =  0  =  n,  ....................................  (26), 

from  which  it  appears  that  there  is  no  horizontal  shear  in  the  steel, 
and  no  twisting  moment  in  vertical  planes  perpendicular  to  x  or  y. 
This  would  be  otherwise  evident  from  considerations  of  symmetry. 
It  will  be  shown  that  this  is  not  true  of  all  other  vertical  planes. 

Again  from  (15)  and  (23)  we  have 

m,  =  h  1  [3*2  -  a2  +  K(3y2  -  62)  ] 
m2=i3[*(3*2-a2)+3V2-&2r 

in  which  we  have  omitted  the  sign  +.  as  superfluous. 

From  (9)  by  help  of  (26)  and  (27),  we  have 

5m!  5m2 

-si  =  --  =  \  q  x,     and  —  s2  =—   -  =  i  q  y  ......  (28) 

ox  oy 

from  which  it  appears  that  any  strip  of  the  panel  parallel  to  x  or  y, 
and  one  unit  wide  exerts  a  shear  at  its  ends  such  as  it  would  if  it  were 
an  isolated  beam  loaded  uniformly  with  an  intensity  of  %q  per  unit 
of  length.  According  to  this,  a  total  shear  of  q  a  b,  which  is  one 
fourth  of  the  total  load  carried  by  the  panel,  appears  at  each  edge  of 
the  panel,  this  total  shear  on  each  edge  being  uniformly  distributed 
along  it. 

It  is  seen  therefore  that  the  form  of  solution  which  we  are 
investigating  implicitly  assumes  that  at  each  edge  of  the  panel  there 
is  some  auxiliary  form  of  structure  that  will  bear  the  shears  coming 
to  it  from  each  side  and  at  the  same  time  assume  the  curvatures  and 
deflections  contemplated  in  (21).  This  will  immediately  engage 
our  further  attention. 


SIDE    BELTS  21 

8.  In  order  to  investigate  more  fully  the  deflections,  stresses 
and  strains  in  the  side  belts  of  any  panel  directly  between  the  mush- 
room heads,  let  us  consider  the  results  just  reached  somewhat  more 
fully. 

The  conclusion  drawn  from  (28)  was,  that  a  panel  with  rein- 
forcement distributed  with  perfect  uniformity  thruout  would  require 
to  be  supported  by  a  narrow  auxiliary  girder  extending  from  column 
to  column  along  each  side,  and  of  such  resisting  moment  as  to  take 
on,  under  its  load,  the  precise  curvature  required  by  the  neutral 
surface  in  (21),  which  curvature  must  be  produced  by  a  uniformly 
distributed  load  of  2  q  a  6,  one  half  of  it  coming  from  each  of  the  two- 
panels  beside  it. 

It  seems  then,  that  up  to  this  point,  we  have  in  reality  been 
treating  the  theory  of  the  continuous  uniform  slab  with  specially 
designed  continuous  beams  supporting  its  edges,  without  as  yet 
investigating  those  beams  in  detail.  But  since  no  such  beams  in 
fact  exist  under  the  flat  slab,  it  is  clear  that  the  side  belts  of  the  slab 
lying  directly  between  the  extended  heads  of  the  columns  must 
discharge  the  functions  which  would  be  discharged  by  the  auxiliary 
beams  just  spoken  of.  Such  functions  must  necessarily  be  added 
to  those  already  discharged  by  those  belts  in  supporting  the  loading 
which  rests  directly  upon  them.  In  order  that  this  may  occur  in  a 
manner  readily  amenable  to  analysis,  the  extended  stiffened  head- 
ings of  the  columns  which  constitute  the  mushrooms  should  in 
general  be  approximately  of  the  diameter  required  to  support  the 
ends  of  a  belt  of  reinforcing  rods  forming  a  flat  beam  which  fills  the 
width  along  the  edge  of  two  adjacent  panels  between  the  two  lines 
of  contra-flexure  on  each  side  of  that  edge,  as  given  in  (24). 

This  requires  that  the  mushroom  head  should  have  a  width  of 
at  least  (1  —  J  V^3 )  =  .423  of  the  width  of  the  slab  between  col- 
umns. For  reasons  that  will  appear  later,  it  is  current  practice  to 
make  these  heads  not  less  than  •&  =  .437  of  this  width. 

The  lines  of  contra-flexure  in  (24)  have  a  fixity  of  position,  (in  a 
flat  slab  constructed  with  mushroom  heads  of  this  size  and  stiffness,) 
under  single  panel  loads,  that  does  not  exist  in  a  uniform  slab,  or 
where  the  headings  are  not  so  stiff.  It  may  be  readily  shown  by 
Mohr's  theorem  respecting  deflection  curves  as  second  moment 
polygons,  that  where  there  are  large  sudden  changes  in  the  magni- 
tude of  the  moment  of  inertia  7,  such  as  exist  in  this  case  at  the  lines 
of  contra-flexure  at  the  edges  of  the  mushroom,  the  lines  of  contra- 
flexure  remain  fixed.  But  in  systems  where  the  diameter  of  the  head 
is  smaller  than  given  above,  or  its  stiffness  is  much  reduced,  these 


22  SIDE    BELTS 

lines  may  be  removed  to  greater  distances  from  the  center  in  loaded 
panels  surrounded  by  tlibse  not  loaded  than  when  all  are  loaded, 
thereby  increasing  the  deflections  and  stresses  in  a  single  loaded 
panel  over  that  of  a  uniformly  loaded  slab  of  many  panels. 

The  lines  of  contra-flexure  in  (24)  separate  the  slab  into  areas 
which  are  largely  independent  of  each  other,  since  no  bending 
moments  are  propogated  from  one  to  another.  The  only  forces 
crossing  these  lines  of  section  are  the  total  vertical  and  horizontal 
shearing  stresses.  The  horizontal  shears  (which  are  unimportant  so 
far  as  deflections  go)  will  be  considered  later  so  far  as  may  be  necessary, 
but  the  vertical  shears  found  by  (28)  are  of  prime  importance.  Let 
us  then  consider  one  of  these  side  belts. 

In  any  extended  slab  with  its  panels  all  loaded  uniformly 
thruout,  the  vertical  shear  must  vanish  at  all  points  along  sections 
made  by  vertical  planes  thru  the  centers  of  columns  at  each  side  of 
any  panel,  as  appears  by  reason  of  symmetry  of  loads.  Let  the 
edges  of  the  side  belts  be  situated  at  some  given  distances,  say  Xi  and 
yi  on  each  side  of  the  centers  of  all  the  panels,  where  x\  and  yv  are 
not  necessarily  the  values  of  x  and  y  given  in  (24),  altho  those 
values  are  also  included  in  this  supposition.  Then  by  (28)  there  is 
a  uniformly  distributed  vertical  shear  of  intensity  \  q_  y\  along  the 
edge  of  the  belt  at  y  =  yi,  even  tho  the  reinforcement  in  the  side 
belt  may  be  greater  than  that  in  the  central  rectangle,  for  the  devia- 
tions caused  by  the  irregularity  of  .its  distribution  may  be  regarded 
as  unimportant  and  practicably  negligible. 

It  may  then  be  assumed  that  any  side  belt  parallel  to  x  must 
carry,  in  addition  to  that  already  provided  for  in  (21),  a  total  loading 
of  q  yi  per  unit  of  length,  uniformly  distributed  along  the  two  edges 
that  are  parallel  to  x.  Now  since  the  width  of  this  belt  is  2(6  —  2/1), 
the  load  already  provided  for  in  (21)  is  \q  per  unit  of  area,  or  q(b-yi) 
per  unit  of  length  parallel  to  x,  which  added  to  that  arising  from  the 
shears  just  mentioned  makes  a  sum  total  of  q  6  per  unit  of  length  of 
belt,  which  it  will  be  noticed  is  independent  of  the  width  of  the  belt. 
In  other  words,  any  such  belt  must  support  a  load  of  one  fourth  of 
the  total  load  on  the  two  panels  of  which  it  forms  a  part,  or  one  half 
of  all  that  lies  between  the  panel  center  lines  which  are  parallel  to 
it  on  either  side.  This  in  effect  transfers  the  entire  loading  of  the 
slab  to  the  side  belts  by  the  agency  of  the  shearing  stresses.  It  does 
this  in  such  a  way  that  one  half  of  the  total  loading  of  the  entire  slab 
is  carried  by  one  set  of  side  belts,  and  the  other  half  by  a  second  set 
which  crosses  the  first  at  right  angles. 


DEFLECTION   OF    SIDE    BELTS  23 

In  those  parts  of  the  slab  area  where  these  sets  of  belts  cross, 
forming  the  heading  of  the  columns,  the  loading  is  superposed  also. 

The  preceeding  investigation  of  the  shears  at  the  edges  of  side 
belts  and  their  loading  is  independent  of  their  width  and  of  the  posi- 
tion of  the  lines  of  contra-flexure,  but  their  width  will  be  assumed  in 
what  follows  to  be  determined  by  the  position  of  those  lines  as  shown 
in  (24)  on  account  of  the  independence  of  action  of  belts  of  their 
width,  as  previously  explained,  where  it  was  shown  that  no  bending 
moments  are  propogated  across  those  lines. 

The  question  now  arises,  how  the  vertical  shears  at  the  edges 
of  the  side  belts  are  distributed  across  their  width  and  carried  by 
them.  Since  by  symmetry  of  loading,  etc.,  there  is  no  vertical 
shear  at  the  edge  of  the  panel  where  y  =  6,  the  shear  must  diminish 
from  each  edge  of  a  belt  to  zero  at  that  line.  If  it  be  assumed  to 
diminish  uniformly,  that  is  equivalent  in  its  action  to  a  uniformly 
distributed  load  on  the  belt,  which  may  be  assumed  in  computation 
to  replace  the  shears  at  the  edges.  Whether  it  will  be  so  distributed 
or  not  depends  upon  the  stiffness  of  the  mushroom  head  and  the 
smallness  of  its  flexure.  Extensometer  measurements  on  the  rods 
of  the  side  belt  of  the  floor  slab  of  the  St.  Paul  Bread  Company 
Building  by  Prof.  Wm.  H.  Kavanaugh  show  beyond  question  that 
in  the  mushroom  system  the  load  is  so  distributed.  Other  exten- 
someter  measurements  to  which  the  writer  has  access  also  show  that 
in  systems  in  which  the  heading  of  the  column  is  not  so  stiff  as  this 
the  distribution  of  loading  cannot  be  taken  as  uniform  over  the  side 
belts. 

Now  the  belt  parallel  to  x  was  shown  to  carry  a  load  per  unit  of 
length  of  q  b  and  to  have  a  width  2(b  —  yi),  in  general,  or  a  width 
26(1  — 1^3)  for  the  belt  between  the  lines  of  contraflexure;  hence 
the  intensity  of  the  loading  on  this  belt  is  q  6/2(6 — yi),  instead  of 
q,  as  it  would  be  in  a  uniformly  loaded  panel  duly  supported  at  its 
edges  by  beams  from  column  to  column.  Let  2  A,  designate  the 
area  of  the  effective  right  cross  section  of  the  steel  in  the  entire  width 
of  a  side  belt  regarded  as  forming  a  single  sheet  of  metal  of  the  width 
of  the  belt;  then  '2A/2(b —  y\)  is  the  effective  right  cross  section 
per  unit  of  width  of  belt,  and  we  may  write  (14)  in  the  form 

I  =  ij  d2  2 A  / 2(6  —  yi) (29) 

We  shall  consider  admissible  values  of  2 A  later. 

Since  the  deflection  of  the  side  belts  may  be  taken  independently 
of  the  rest  of  the  slab,  let  those  values  for  the  intensity  of  loading 
.and  the  moment  of  inertia  (29)  be  introduced  into  (21). 


24  STRESSES   IN   SIDE    BELTS 

We  then  obtain  an  expression  for  the  law  governing  the  deflec- 
tion of  that  part  of  the  side  belts  parallel  to  x  which  lies  between  the 
mushroom  heads,  and  is  bounded  by  lines  of  contra-flexure,  viz  : 


Z= 


48  E  i  j  d2 


with  a  corresponding  equation  for  the  side  belts  parallel  to  y,  which 
may  be  obtained  by  replacing  q  b  in  (30)  by  q  a.  Call  this  second 
equation  (31).  Now  (30)  and  (31)  would  hold  thruout  the  entire 
length  of  these  belts  from  column  to  column  were  they  entirely 
separate  from  each  other  and  from  the  diagonal  belts  where  they 
cross  each  other.  It  will  be  necessary  later  to  obtain  the  equation 
which  holds  true  where  these  belts  cross  and  combine  with  each  other. 

9.  Practical  formulas  for  the  stresses  in  the  steel  and  concrete 
of  side  belts  between  the  lines  of  contra-flexure  will  now  be  obtained 
from  (30)  and  (31). 

In  order  to  do  this,  consider  the  summation  in  (30)  expressing 
the  effective  cross  section  of  the  steel  in  the  mid  area  of  the  side  belt 
regarded  as  forming  a  single  uniform  sheet,  that  mid  area  being 
bounded  on  all  sides  by  lines  of  contra-flexure. 

It  is  to  be  noticed  that  the  factor  (1  —  K2)  of  (30)  takes  into 
account  the  fact  that  the  lattice  of  rods  forming  the  reinforcement  is 
less  effective  than  the  same  amount  of  metal  in  the  form  of  a  sheet, 
the  only  question  left  being  this:  Will  the  great  irregularity  of 
distribution  of  the  reinforcement  in  this  area  cause  it  to  act  differ- 
ently to  any  noticeable  extent  from  the  manner  in  which  the  same 
amount  of  metal  would  act  were  it  possible  to  distribute  it  uniformly 
over  the  entire  area?  There  are  strong  reasons  which  go  to  sustain 
the  view  that  this  irregularity  of  distribution  is  negligible  in  the 
standard  mushroom  slab,  at  least  for  loads  less  than  those  that 
stress  the  steel  below  the  yield  point,  or  do  not  stress  the  concrete 
for  too  long  a  time  while  it  is  imperfectly  cured.  On  examining  a 
diagram  of  the  reinforcing  rods  of  a  slab  made  with  square  panels 
of  such  proportions  that  the  width  of  the  belts  is  one  half  the  distance 
between  columns,  then  the  pattern  previously  mentioned  into  which 
it  would  be  divided  by  these  belts  will  be  seen  to  consist  of  equal 
squares  whose  edges  are  equal  to  the  width  of  the  belts,  with  one 
central  square  in  each  panel  concave  upwards,  and  one  half  of  each 
of  the  saddle  shaped  squares  which  border  it,  also  lying  within  the 
same  panel,  and  one  quarter  of  each  of  the  four  convex  squares  at  the 


MEAN   REINFORCEMENT   OF    SIDE    BELT  25 

head  of  each  of  the  columns  at  the  corners  of  the  panel,  also  lying 
within  the  same  panel,  see  Fig.  3,  page  7. 

Each  side  square  will  be  found  in  this  case  to  have  double  (or 
two  belt)  reinforcement  over  one  half  or  its  area,  single  belt  rein- 
forcement over  a  diamond  occupying  one  fourth  of  its  mid  area,  and 
triple  reinforcement  over  four  triangular  areas  along  its  sides  which 
together  cover  one  fourth  of  the  square.  This  gives  a  mean  value 
of  2 A  =  2  Ai'm  which  AI  is  the  total  right  cross  section  of  the  rods 
in  the  side  belt. 

The  belts  in  the  standard  mushroom  are,  however,  not  so  wide 
as  this,  since  that  system  simply  requires  that  the  edges  of  the  side 
and  diagonal  belts  intersect  in  a  single  point,  Fig.  2,  instead  of  forming 
four  areas  of  triple  reinforcement  on  the  sides.  This  makes  the 
width  of  the  singly  reinforced  diamond  sufficient  to  just  reach  across 
the  side  belt.  In  this  practical  case  we  find  that  very  approximately 

2A  =  1.5  Al (32) 

in  which,  as  before,  A!  is  the  total  right  cross  section  of  the  side  belt 
in  square  inches.  It  is  evidently  impossible  for  this  single  side  belt 
of  rods  which  crosses  the  diamond,  to  elongate  without  a  correspond- 
ing equal  elongation  of  the  double  reinforcement  on  all  its  sides,  or 
at  least  it  is  impossible  for  readjustments  to  take  place  in  any  short 
time  such  as  will  make  these  direct  deformations  within  the  diamond 
larger  than  those  in  the  areas  along  side  of  it,  or  before  somewhat 
more  permanent  deformations  have  taken  place  in  the  concrete. 

In  cases  where  the  column  heads  are  smaller  than  the  standard, 
and  the  side  belts  still  narrower,  not  only  may  S A  become  much 
less  than  1.5  A  i  but  the  belt  become  so  weakened  near  the  central 
diamond  as  to  render  it  very  questionable  whether  the  irregularity 
of  distribution  of  steel  in  the  area  considered  may  be  safely  disre- 
garded. Diminution  of  the  size  of  the  heading  thus  not  only  dim- 
inishes cantilever  action,  but  reduces  the  effective  resistance  of  the 
reinforcing  steel.  Not  much  diminution  of  the  size  of  head  would 
be  required  to  reduce  the  value  of  SA  to  an  amount  as  small  as  AI. 

Introducing  the  estimate  given  in  (32)  for  the  standard  mush- 
room into  (30)  we  derive  by  (23),  (23o)  and  (3), for  that  part  of  the 
side  belt  parallel  to  x  between  x  =  +  |  aV^3  and  x  =  —  ^  aVs, 

^d2  z         ,    (l-K2)gb 

—  (3z  —  a2) 


5  x 


MI  =  1.5  Av  j  dfs  =  ±  -  q  b  (3x2  —  a2) 

12 


(33) 


26  TRUE    STRESSES    IN    SIDE    BELT 

in  which  MI  is  the  total  true  moment  of  resistance  of  the  side  belt, 
4  is  the  true  stress  per  square  unit  of  the  reinforcement  in  the  side 
belt,  and  1.5  A  i  is  the  effective  right  cross  section  of  the  reinforce- 
ment. This  is  independent  of  y  as  before  noted,  showing  that  the 
values  of  /s  and  ei  are  the  same  for  one  rod  as  for  another,  but  they 
attain  their  greatest  values  at  the  mid  length  where  x  =  0.  If  units 
be  pounds  and  inches,  and  we  assume  j  =  0.91  for  the  very  small 
percentage  of  reinforcement  of  the  standard  mushroom  system,  then 
by  (33)  and  (6)  the  practical  formulas  for  design  are: 

3  q  a2  b  W  L 

Js    = 


4  x  18  x  0.91  di  A1       175  di  Al 
Mi  =  1.5AiM/s  = 


(34) 
W  L 


128 

in  which  /s  is  the  true  stress  in  the  steel,  and  MI  is  the  true  bending 
moment  of  the  effective  cross  section  1.5  A\  of  the  steel  in  the  entire 
belt  as  shown  by  the  elongation  (at  mid  span)  of  the  rods  in  a  side 
belt  of  length  L,  where  L  is  either  2a  or  26,  and  W  =  4  q  a  b  is  the 
total  load  on  the  panel  in  pounds,  where  d\  is  the  vertical  distance 
from  the  center  of  the  rods  in  the  single  belt  at  mid  span  to  the  top 
surface  of  the  slab. 

While  the  values  obtained  from  (34)  are  conservative  forj  =  0.91, 
corresponding  to  a  percentage  of  reinforcement  for  one  belt  of  less 
than  0.25%,  (34)  should  be  regarded  merely  in  the  light  of  a  speci- 
men equation  for  that  percentage,  and  any  slab  where  the  percent- 
age differs  materially  from  that  assumed  value  should  be  submitted 
to  separate  computation  in  the  same  manner. 

Values  of  j  are  given  for  beams  by  Turnearue  and  Maurer  in 
their  "  Reinforced  Concrete  Construction,"  page  57,  for  different 
percentages  of  reinforcement  on  the  straight  line  theory,  which 
latter  is  now  accepted  usage.  As  already  stated,  standard  mush- 
room design  makes  the  percentage  of  reinforcement  for  warehouse 
floors  where  the  panels  are,  say  20'  x  20',  as  low  as  0.25%  or  less,  at 
the  middle  of  the  side  belts,  reckoned  on  the  beam  theory.  But  in 
heavier  and  larger  construction  it  may  reach  0.33%. 

We  have  taken  the  mean  available  steel  in  the  belt  as  1.5  A  b 
hence  the  mean  slab  reinforcement  will  not  be  less  than  1.5  x  0.23  = 
0.4%  in  the  side  belt  areas  between  lines  of  contra-flexure. 

In  case  we  assume  the  ratio  of  Es  for  steel  to  E0  for  concrete  to 
be  15,  as  is  often  prescribed,  we  find  the  above  stated  value  of  j  as  a 


THEORETICAL   AND    EXPERIMENTAL    STRESSES    COMPARED  27 

good  mean  value,  which  will  be  less  in  cases  where  the  percentage 
of  steel  is  greater.  The  small  percentage  of  steel  and  great  relative 
thickness  of  concrete  is  one  of  the  distinguishing  features  of  the 
standard  mushroom  design. 

We  may  write  (34)  in  the  form : 

W  L  W  L 

fs=~  -}SindMl=Aljdfs  =  —      -....  (34a) 

175  d1  A!  192 

in  which  M(  is  the  true  bending  moment  of  the  actual  cross  section 
AI  at  mid  belt.  We  have  written  this  modification  of  (34), not  for 
use  in  design,  but  merely  for  the  purpose  of  instituting  a  comparison 
with  empirical  formulas  obtained  by  Mr.  Turner  to  express  the 
results  of  numerous  tests  made  by  him.  On  pages  26  and  28  of  his 
" Concrete  Steel  Construction"  he  has  given  equations  expressing 
the  values  of  stresses  and  moments  in  mushroom  slabs  which  in  our 
notation  may  be  written  as  follows: — 

WL  W  L  W  L 

M!  =  Alj  dfa  =  -       -  ,  and/s  =  -  -  .  (35) 

200  200  x  0.85  d  Al      170  d  A, 

in  which  he  has  assumed  0.85  as  a  mean  value  of  j. 

It  is  seen  that  equations  (34a),  obtained  from  rational  theory 
alone,  are  in  practical  agreement  with  (35),  which  were  deduced 
from  experimental  tests  of  mushroom  slabs,  where  the  numerical 
coefficient  introduced  is  entirely  empirical. 

As  will  be  seen  later,  (34)  is  the  equation  which  ultimately  con- 
trols the  design  of  the  slab  reinforcement;  so  that  the  agreement  of 
these  two  entirely  independent  methods  of  establishing  this  funda- 
mental equation  cannot  but  be  regarded  with  great  satisfaction  as 
affording  a  secure  basis  for  designs  that  may  be  safely  guaranteed 
by  the  constructor,  as  has  been  the  custom  in  constructing  standard 
mushroom  slabs. 

The  slab  theory  here  put  forth  diverges  so  radically  from  the 
results  of  beam  theory  that  we  introduce  here  the  following  compar- 
ative computation  of  the  smallest  values  of  true  bending  moment 
and  stress  in  steel,  which  can  be  obtained  by  beam  theory  for  the 
side  belt  parallel  to  x,  as  follows: — 

That  part  of  the  side  belt  between  the  lines  of  contra-flexure  is 
simply  supported  at  its  ends  by  shearing  stresses,  and  so  may  be 
taken  to  be  a  simple  beam  resting  on  supports  at  these  end  lines. 


28  STRESSES    BY    BEAM    THEORY    AND    BY   TEST 

Hence  the  true  stress  /s  and  the  true  bending  moment  M '  at  the 
middle  of  this  simple  uniformly  loaded  beam  may  be  computed  from 
the  equation, 

M'  =  A,jdfs  =  i  W'  L' (36) 

in  which  M '  is  the  total  moment  of  resistance. 

AI  is  the  total  right  cross  section  of  the  reinforcement,  W'  is  the 
total  uniformly  distributed  load,  and  L'  is  the  length  of  the  beam. 
The  length  of  the  simple  beam  in  that  case  is  evidently  the  distance 
along  x  between  lines  of  contra-flexure,  viz,  L'  =  f  a^3  =  J  L  V^3, 
where  L  is  the  edge  of  the  panel,  and  the  total  load  at  most  will  be 
that  already  proven  to  be  carried  by  the  side  belt  viz,  qb  per  unit  of 
length,  or  a  total  for  a  span  L'of  W'  =  qb  L'  =  f  qab  ^3  =  £  TF  V^3 
where  W  =  4  qab  is  the  total  load  on  the  panel,  hence 

W  L 

M    =  A1jdfs  =        - (36a) 

48 

It  thus  appears  that  according  to  simple  beam  theory  the  true 
stress,  or  the  cross  section  of  steel  required  in  the  belt,  is  four  times 
that  obtained  by  slab  theory  as  shown  by  (34a).  Since  (34a)  is  in 
good  accord  with  experimental  tests,  this  comparison  justifies  the 
statements  made  near  the  beginning  of  this  paper  respecting  the 
inapplicability  of  beam  theory  to  the  computation  of  slab  design. 

The  floor  of  the  St.  Paul  Bread  Co.  Building,  previously  men- 
tioned,is  a  rough  slab  6"  thick,  and  has  panels  16'  x  15',  with  ten 
3/8"  round  rod  reinforcement  in  each  belt,  built  for  a  design  load  of  100 
pounds  per  square  foot;  constructed  in  winter  and  frozen,  the  final 
test  was  not  made  until  the  end  of  its  first  summer  after  unusually 
complete  curing,  such  as  might  make  the  value  of  K  given  in  (6) 
not  entirely  applicable.  In  one  long  side  belt,  extensometer  measure- 
ments were  made  at  the  mid  span  on  three  rods,  (1)  a  middle  rod, 
(2)  an  intermediate  rod  and  (3)  an  outside  rod  of  the  belt,  with  the 
following  results  for  the  given  live  load  in  pounds  per  square  foot: 

Live  Loads  108 . 4  #  316 . 8  #  416 . 8  # 


/.  -  E  ei  (1) 

(2) 
(3) 

7650 
7080 
7320 

15000 
14190 
13920 

17940 
16470 
17160 

Average 

7350 

14370 

17200 

A  by  (34) 

5000 

14440 

19000 

STRESSES   AT   YIELD    POINT  29 

The  observed  results  are  seen  to  be  in  excellent  agreement  with 
those  computed  from  (34)  for  the  heavier  loads,  while  any  disagree- 
ment is  on  the  safe  side.  Agreement  is  not  expected  for  light  loads. 

The  accuracy  and  applicability  of  (34)  and  preceeding  formulas 
is  dependent  on  the  fixity  of  the  lines  of  contra-flexure  (24)  which 
were  previously  stated  to  be  practically  immovable  because  of  the 
sudden  large  change  of  the  moment  of  resistance  of  the  slab  at  those 
lines.  That  fact  may  be  put  in  a  more  definite  and  convincing 
form  than  has  been  done  so  far.  Consider  for  a  moment  that  form 
of  continuous  cantilever  bridge  where  there  are  joints  between  the 
cantilevers  over  the  successive  piers  (which  are  in  the  form  of  a  letter 
T)  and  the  intermediate  short  spans  which  connect  the  extremities 
of  the  cantilevers.  At  such  joints  the  resisting  moments  vanish,  and 
they  form  in  a  sense  artificially  fixed  points  of  contra-flexure.  The 
same  thing  approximately  occurs  at  the  edge  of  the  mushroom, 
because  there  the  reinforcing  steel  rapidly  dips  down  from  a  level 
above  the  neutral  plane  to  one  below  it,  and  the  sign  of  the  moment 
of  resistance  changes  thru  zero  at  that  edge. 

Furthermore,  it  may  be  proper  to  state  in  this  connection  that 
the  foregoing  theory  has  been  developed  in  consonance  with  the 
general  principles  of  elasticity,  and  that  somewhat  different  condi- 
tions and  relations  are  thought  to  exist  when  the  steel  at  the  middle 
of  the  side  belts  reaches  its  yield  point,  as  it  does  in  advance  of  the 
rest  of  the  reinforcement.  As  the  yield  point  is  reached  equations 
(34)  no  longer  hold;  for,  as  will  be  seen  more  clearly  later,  the  single 
belt  of  reinforcing  steel,  which  crosses  the  circumference  of  an  ap- 
proximately circular  area  of  radius  L  /  2  about  the  center  of  each 
column,  will  everywhere  reach  the  yield  point  at  practically  the  same 
instant,  and  if  loaded  much  beyond  this  will  develop  a  continuous 
line  of  weakness  there.  The  equations  that  hold  in  this  case  will  be 
approximately  those  due  to  the  actual  cross  section  A!  of  the  belt,  in 
place  of  (34),  which  contain  the  effective  cross  section,  viz: 

3  q  a2  b  W  L 

fs    = 


4x  12  x  0.91  dl  AI        117  di  A 


•  -  (37) 
W  L 


128 

which  may  be  regarded  as  expressing  the  relations  that  exist  at  the 
limit  of  the  elastic  strength  of  the  slab  and  the  beginning  of  perma- 
nent deformation,  tho  not  necessarily  of  collapse. 


30  STRESSES   IN   CONCRETE 

The  percentage  of  reinforcement  in  standard  mushroom  slabs 
is  small  enough  to  make  their  elastic  properties  depend  upon 
the  resistance  of  the  steel.  The  stresses  in  the  concrete  may  then  be 
be  computed  from  those  in  the  steel,  but  many  uncertainties  attend 
any  such  computation.  It  is  usage,  fixed  by  the  ordinances  of  the 
building  codes  of  most  cities  to  require  the  application  of  the  so 
called  " straight  line  theory"  in  such  computations,  not  because  that 
will  give  results  which  will  be  verified  by  extensometer  tests  of  com- 
pressions in  the  concrete,  for  it  will  not,  but  because  it  is  definite 
and  on  the  side  of  safety.  Furthermore  it  is  usually  prescribed 
that  the  ratio  of  the  modulus  of  elasticity  of  steel  divided  by  that 
of  concrete  shall  be  assumed  to  be  15,  where  the  moduli  are  unknown 
by  actual  test  of  the  materials.  This  is  usually  far  from  a  correct 
value.  The  consequence  is  that  the  results  of  computation  of  the 
stresses  in  concrete  are  highly  artificial  in  character,  and  should  not 
be  expected  to  be  in  agreement  with  extensometer  tests.  With  this 
understanding  the  computed  stress  in  the  concrete  at  the  middle  of 
the  side  belt  will  be  found  as  follows: — 

Let  id  be  the  distance  from  the  center  of  the  steel  to  the  neutral 
plane.  (It  happens  to  be  more  convenient  in  this  investigation  to 
use  this  distance  id  here  and  in  our  previous  formulas  than  to  intro- 
duce the  distance  from  the  neutral  axis  of  the  slab  to  the  compressed 
surface  of  the  concrete,  as  is  done  by  many  writers,  under  the  desig- 
nation k  d.  These  quantities  are  so  related  that  i  +  k  =  1  ). 

Then,  as  is  well  known  from  the  geometry  of  the  flexure  of 
reinforced  concrete  beams,  in  case  tension  of  concrete  is  disregarded, 

k     Ec 
/c  -  ~-'-fs (38) 

i      Es 

where  the  subscripts  c    and    s    refer    to    concrete    and    to    steel 
respectively. 

Applying  (38)  to  the  greatest  computed  stress  fa  =  19000  in 
the  St.  Paul  Bread  Go's  Building,  gives  a  computed  stress /c  =  492; 
but  taking  the  greatest  observed  stress  fa  =  17940gives/c  =  465  Ibs. 
per  sq.  inch,  as  the  greatest  computed  compressive  stress  in  the 
concrete  at  the  middle  of  the  side  belt,  if  i  =  0,72. 

The  tensile  stress  across  the  middle  of  the  side  belt  at  the 
extreme  fiber  of  its  upper  surface  is  fixed  by  the  curvature  of  the 
vertical  sections  of  the  slab  in  planes  that  cut  the  side  belt  at  right 
angles.  As  stated  previously  all  such  planes  make  cross  sections  of 
the  side  belt  that  are  identical  in  shape.  That  is  a  consequence  of 


DEFLECTIONS  IN  COLUMN  HEAD  AREAS  31 

the  conclusion  reached  previously,  that  all  the  rods  in  the  side  belt 
are  subjected  to  equal  tensions.  The  curvature  of  these  sections 
is  controlled  by  the  stiffness  of  the  mushroom  heads,  which  is  so 
great  as  to  make  the  curvature  very  small.  No  considerable  tensile 
cross  stresses  are  consequently  to  be  apprehended;  but  in  case  the 
stiffness  of  the  head  were  to  be  decreased,  stresses  might  arise  such 
as  to  develop  longitudinal  cracks  over  the  middle  rod  of  the  side  belts. 

10.  In  order  to  obtain  practical  formulas  for  the  deflections  and 
stresses  in  the  steel  thruout  the  areas  at  and  near  the  tops  of  the 
columns  where  all  the  belts  cross  each  other,  and  lying  between  lines 
of  contra-flexure,  we  shall  have  recourse  to  (30)  and  (31)  which  are 
here  superimposed  on  each  other,  and  combined  together.  Were 
there  no  steel  here  in  addition  to  the  side  belts,  that  superposition 
could  be  correctly  effected  by  writing  a  value  of  z  whose  numerator 
would  be  the  sum  of  the  numerators  of  (30)  and  (31),  for  that  would 
superpose  the  loads  of  the  two  side  belts,  and  thus  place  the  total 
required  loading  upon  this  area  as  previously  explained;  and  then  by 
writing  for  a  denominator  the  sum  of  the  denominators  of  (30)  and 
(31),  for  that  would  superpose  and  combine  the  resistance  of  all  the 
steel  in  both  belts.  But  such  a  result  would  leave  out  of  account  the 
reinforcement  arising  from  the  diagonal  rods,  and  the  radial  and 
ring  rods,  which  should  also  be  reckoned  in  as  furnishing  part  of  the 
resistance. 

Supposing  this  additional  steel  to  be  distributed  in  this  area  in 
the  same  manner  as  is  that  of  the  side  belts,  a  supposition  which  is 
very  close  to  the  fact,  we  may  write 

..(-*>.<>+») 

48  Eijd2  2  A 

in  which  1<A  is  the  cross  section  of  the  total  reinforcement  in  this 
area  regarded  as  forming  a  uniform  sheet,  i  and  j  stand  for  mean 
values  that  have  to  be  determined  by  the  percentage  of  reinforce- 
ment and  its  position,  while  d  is  the  mean  distance  of  the  center  of 
action  of  the  steel » above  the  lower  compressed  surface  of  the  con- 
crete at  the  point  xy. 

We  may  conservatively  assume  in  the  standard  mushroom 
that  the  center  of  action  of  the  steel  is  at  the  center  of  the  third  layer 
of  rods  from  the  top,  as  will  appear  more  clearly  later.  This  defines 
d,  which  we  shall  consequently  designate  by  d3. 

It  remains  therefore  to  estimate  the  amount  of  the  total  rein- 
forcement SA,  and  then  find  mean  values  of  i  and  j. 


32  MEAN   REINFORCEMENT   OF    HEAD 

In  case  of  reinforcing  rods  which  are  all  of  them  continuous 
over  a  head  without  laps,  the  percentage  of  reinforcement  falls  only 
slightly  below  4  times  that  at  the  middle  of  a  side  belt;  but  on  the 
other  hand  were  none  of  them  continuous  for  more  than  one  panel 
and  each  lap  reached  beyond  the  center  of  the  column  to  the  edge 
of  the  mushroom,  the  percentage  of  reinforcement  would  not  be  less 
than  7  times  that  at  the  middle  of  a  side  belt,  and  to  this  must  be 
added  that  due  to  the  steel  in  the  radial  and  ring  rods.  Thus  the 
percentage  of  reinforcement  here  may  be  varied  not  only  by  reason 
of  the  larger  or  smaller  number  of  laps  over  each  mushroom,  but  by 
reason  of  the  length  of  the  laps,  from  perhaps  3.75  to  7  times  that 
at  the  middle  of  a  side  belt.  For  standard  mushroom  construction 
using  long  rods,  it  may  be  taken  conservatively  as  a  4.25  times  that 
at  the  middle  of  a  side  belt. 

It  is  impossible  to  make  an  estimate  that  will  be  accurate  for 
all  cases,  but  commonly  the  8  radial  rods  of  a  20'  x  20'  panel  are 
equivalent  in  amount  to  a  single  1-1/8"  round  rod,  or  a  1"  square 
bar  circumscribing  the  area  under  consideration,  that  is  to  4  square 
inches  of  additional  reinforcement  to  be  distributed  in  the  width  of 
a  single  side  belt. 

The  two  rings  rod,  of  which  the  larger  is  commonly  7/8"  round, 
and  the  smaller  5/8"  round,  may  be  taken  to  increase  the  reinforce- 
ment of  this  area  by  at  least  one  square  inch  of  cross  section,  giving 
all  told  some  five  square  inches  of  cross  section  additional,  equiva- 
lent forty-five  3/8"  round  rods,  or  twenty-one  1/2"  rods.  It  thus 
appears  that  the  increased  reinforcement  from  this  source  reaches 
from  2  to  4  times  A^  and  we  may  safely  assume  a  mean  total  rein- 
forcement over  this  area  of 

24  =  7.5  AI (40) 

of  which  the  center  of  action  may  be  pretty  accurately  stated  to  be 
at  the  middle  of  the  third  layer  of  reinforcement  rods  from  the  top. 

In  the  standard  design  of  mushroom  floors  for  warehouses  with 
panels  about  20'  x  20',  the  mean  percentage  of  reinforcement  for  a 
single  belt  AI  being  about  0.23%,  may  be  taken  by  (40)  for  a  rein- 
forcement 7.5  AI  as 

7.5  x  0.23  +   =   1.75%     The  corresponding  value  of  j  is  0.83, 

and  we  shall  have 

j  S  A  =  0.83  x  7.5  Al  =  QAl (41) 

As  previously  stated,  these  equations  (containing  estimated  mean 
numerical  values)  are  given  as  a  specimen  computation  for  the  purpose 


STRESSES    AT   EDGE    OF    CAP  33 

of  making  comparisons.  In  actual  design,  computations  like  these 
should  be  made  which  introduce  the  exact  values  appearing  in  the 
design  under  consideration. 

We  now  derive  from  (39)  and  (40)  by  the  help  of  (23)  the  follow- 
ing equations  for  this  area  where  the  belts  all  cross: — 

/s  =  Ed  =  ±Eid3-  —  (3z2  —  a2) 

ox  90  j '  d3  A  i 

d-K2)  M42) 

Mi  =  7.5  Ai  j  d3fs  =  q  (a  +  b)  (3x2  —  a2) 

12 

in  which  j  and  d3  are  less  than  in  (33)  and  (34),  as  has  been  stated 
previously. 

Apply  (42)  to  find  the  stresses  at  the  edge  of  the  column   cap 
on  the  long  side  LI. 

Let  B  =  2x  be  the  shortest  distance  along  the  middle  of  the 
side  belt  parallel  to  x  between  the  edges  of  the  caps  of  two  adjacent 
columns,  and  introduce  the  values  j  =  0.83,  K  =  0.5,  and  W  = 
then; 

W  Li  (Li  +  L2)     (3B2/Ll  —1) 
800  d3  Ai  L2 

WLl  (Li  +  L2)     (3B2/L2  —  :     (' 


in  which  7.5  AI  is  the  effective  cross  section  of  the  steel  in  this  area, 
and  MI  is  the  true  resisting  moment  of  the  steel  derived  from  the 
elongation,  and  d3  is  as  stated  after  (39). 

Take  the  case  of  a  square  panel,  and  assume  the  diameter  of 
the  column  cap  to  be  0.2Li,  then  B  =  O.SLi  and  (43)  reduce  to: 


(44) 
W 


It  will  be  readily  seen  that  if  d3  in  (44)  is  more  than  0.4  of  the  vertical 
distance  designated  by  dv  in  (34),  (as  it  must  be)  then  the  stress  /s 
in  (34)  at  the  middle  of  the  side  belt  exceeds  /s  in  (43)  at  the  edge 


34  GREATEST   STRESSES   OVER    COLUMN 

of  the  cap.  But  this  does  not  prove  that  the  stress  in  the  concrete 
at  the  edge  of  the  cap  i's  less  than  that  at  the  middle  of  the  side  belt, 
for,  the  value  of  i  in  (37)  at  the  middle  of  the  side  belt  is  about  2/3 
and  at  the  edge  of  the  cap  about  1/2,  as  will  be  seen  by  consulting 
Turneaure  and  Maurer,  page  57,  for  values  of  i  corresponding  to  the 
values  of  j  at  these  points.  Hence,  using  these  values  of  i,  if  primes 
be  used  to  designate  the  stress  at  the  edge  of  the  cap,  we  have  by 

(38),  f'c/fc  =  2/;  //s (45) 

from  which  it  is  seen  that  the  stress  /s'  at  the  edge  of  the  cap  must 
be  only  half  that  in  the  side  belt  in  order  that  the  corresponding 
stresses  in  the  concrete  may  be  equal.  But  ordinarily  2/s'  >/s,  and 
so  /c'  >/c.  The  stress  in  the  concrete  at  the  edge  of  the  cap  will  be 
computed  from  that  of  the  steel  found  in  (44)  by  using  (38),  in  which 
if  we  put  i  =  K  =  \,  we  have  /c'  =  /s'  /15,  as  the  computed  value 
of  the  stress  at  the  edge  of  the  cap. 

Tests  have  seemed  to  show  that  much  higher  compressive 
stresses  may  be  safely  permitted  in  the  concrete  around  column  caps 
where  there  is  compression  in  two  directions,  than  in  the  extreme 
fiber  of  a  beam  where  compression  takes  place  in  one  direction  only. 
A  like  principle  applied  to  the  extreme  fiber  at  the  middle  of  the 
side  belt  where  tension  exists  at  right  angles  to  the  compression 
would  show  that  there  only  a  low  value  should  be  permitted  in 
compression. 

In  order  to  compare  the  greatest  stresses  in  the  steel  across  the 
mushroom  with  that  at  the  middle  of  the  side  belts  in  a  square  panel 
let  B  =  LI  =  L2  in  (43),  then  the  stress  in  a  section  thru  the  column 
center  along  the  edges  of  the  panel  over  the  mushroom  area  is  found 
from  the  following  equations: 


/s    = 


200  d3  AI 


32 


(46) 


which  are  to  be  compared  with  (34),  from  which  it  appears  that  if 
d3  in  (46)  is  more  than  7/8  of  di  in  (34),  the  stress  in  the  steel  across 
the  mushroom  is  less  than  at  the  center  of  the  side  belts.  In  any 
case  these  stresses  are  so  nearly  equal  that  the  inadvisability  of 
decreasing  the  steel  in  the  mushroom  head  below  standards  indicated 
above  is  evident.  However,  some  of  the  steel  at  the  edge  of  the 
mushroom  especially  the  outer  hoop  is  at  such  level  in  this  right 


STRESSES    OVER    CAP  35 

section  of  the  head  as  possibly  to  assist  the  concrete  in  bearing  com- 
pressive  stresses.  Such  a  large  portion  of  this  section,  moreover, 
falls  within  the  cap,  that  no  question  of  its  stability  and  safety  need 
arise,  in  case  the  collar  band  of  the  column  is  sufficient  to  resist  the 
comparatively  small  tensions  of  the  radial  rods. 

It  will  be  noticed  that  in  order  to  make  fa  and  /c  as  small  as 
possible  in  this  area  d3  must  be  made  as  large  as  possible,  i.  e.,  the 
steel  at  the  edge  of  the  cap  must  be  raised  as  near  the  top  of  the  slab 
as  possible.  Neglect  of  this  is  to  invite  failure  and  weakness  such 
as  has  overtaken  certain  imitators  of  the  mushroom  system. 

A  final  remark  is  here  in  place  respecting  the  values  of  j  and  d% 
in  this  area.  The  stresses  /„  and  /c  diminish  very  rapidly  towards 
the  lines  of  contra-flexure,  where  they  vanish,  and  the  fact  that  the 
steel  also  rapidly  increases  its  distance  from  the  top  of  the  slab  at 
the  same  time  might  be  regarded  at  first  thought  as  requiring  some 
modification  of  the  assumptions  we  have  made  as  to  the  values  of 
j  and  d3,  which  are  approximately  correct  at  the  edge  of  the  cap 
where  the  steel  is  placed  as  near  the  top  surface  as  due  covering  will 
permit.  But  the  fact  is  this:  the  only  consideration  of  importance 
is  the  one  respecting  the  position  of  the  steel  in  that  part  of  this 
area  where  the  moments  and  stresses  are  large.  The  effect  of  the 
position  of  the  steel  near  the  lines  of  contra-flexure  is  negligible,  and 
the  fact  that  the  amount  of  reinforcement  may  be  somewhat  smaller 
near  these  lines  than  elsewhere  may  also  be  neglected,  so  that  the 
mean  effective  reinforcement  previously  estimated  is  likely  to  be  an 
underestimate  rather  than  the  reverse.  Further,  the  fact  that  the 
slab  is  practically  clamped  horizontally  either  at  the  edge  of  the  cap 
or  the  edge  of  the  superposed  column,  instead  of  at  its  center  as 
assumed  in  our  formulas,  renders  the  results  given  thus  far  slightly 
too  large. 

Good  average  values  of  the  size  of  steel  used  in  the  standard 
mushroom  system  of  medium  span  would  make  the  radial  rods 
9/8"  round,  the  outer  ring  rod  7/8"  round,  the  inner  ring  rod 
5/8"  and  the  belt  rods  3/8"  round.  The  importance  of  having 
the  belt  rods  small  is  that  for  a  given  thickness  of  slab  the  smaller 
these  rods  are  the  larger  is  d  in  both  (34)  and  (43)  and  consequently 
the  smaller  is/s  and  A\. 

11.  In  attempting  to  consider  the  stresses  in  the  diagonal  rods 
of  the  central  rectangle  between  the  side  belts  of  a  panel,  it  will  be 
noticed,  as  stated  before,  that  no  true  bending  moments  are  propo- 
gated  across  the  vertical  planes  or  lines  of  contra-flexure  (24)  which 


36  DEFLECTIONS  IN  CENTRAL  AREA  OF  PANEL 

bound  it,  and  since  the.  vertical  shearing  stresses  at  these  lines  are 
uniformly  distributed  along  them,  as  already  shown,  (28),  there  are 
no  true  twisting  moments  in  these  planes.  The  curvatures  of  this 
rectangle  will  consequently  depend  upon  its  own  loading  and  the 
resistance  of  its  own  moment  of  inertia,  regarded  as  uniformly  dis- 
tributed, independently  of  that  of  other  parts  of  the  slab. 

Hence  (21)  may  be  correctly  applied  to  this  area,  regardless  of 
the  values  which  7  (and  q)  may  assume  elsewhere,  provided  only 
that  the  values  of  /  in  other  areas  may  be  assumed  to  have  constant 
values  thruout  those  areas,  and,  further,  that  those  areas  are  sym- 
metrically disposed,  so  that  all  central  rectangles  have  one  and  the 
same  given  value  of  /  thruout,  all  side  belts  also  have  one  given 
value  of  /,  and  the  mushroom  heads  have  a  given  value  also,  each  of 
these  three  sorts  of  areas  being  independent.  The  truth  of  this 
proposition  has  been  heretofore  tacitly  assumed  in  applying  (21) 
to  these  latter  areas  as  has  been  done. 

It  will  be  seen  however,  that  the  values  of  z  obtained  from  such 
diverse  equations  express  deflections  of  any  point  xy  on  the  supposi- 
tion that  all  the  areas  considered  have  the  same  value  of  /;  but  these 
separate  equations,  each  with  its  own  peculiar  value  of  7,  can  be 
used  separately  to  find  the  difference  of  level  Zi  —  z2  between  any 
two  points  Xi  yi  and  x^  y%  which  lie  in  an  area  where  /  may  be  regarded 
as  constant.  We  shall  return  to  this  point  when  we  come  to  the 
derivation  of  practical  deflection  formulas. 

For  convenience  in  computing  stresses  in  the  rods  of  the  diago- 
nal belt,  let  the  direction  of  the  coordinates  be  changed  so  that  in 
square  panels  they  will  lie  along  the  diagonals  which  make  angles 
of  45  °  with  those  used  thus  far.  In  (21)  let 

x  =  %V2(x'  +  y\         y  =  %V2(x'--y'),  then 


22  2 


24  E  i  j  d2±A 


~  a2(x     +  y'2)  +  x'2y2  +  1(*       )]  ....  (47) 


in  which  the  panel  is  square  and  the  axes  of  x'  and  y'  lie  along  its 
diagonals,  while  the  value  of  Z  A  /  g  is  the  effective  cross  section  per 
unit  of  width  of  all  the  reinforcement  in  this  area  regarded  as  a 
single  uniform  sheet  of  metal,  and  g  =  7/8  a,  is  the  width  of  a 
diagonal  belt,  and  is  equal  to  the  diameter  of  the  mushroom  head. 
In  rectangular  panels  g  =  7/16  (a  +  6). 


ELONGATIONS  AND  SHEARS  IN  CENTRAL  AREA  37 

From  (34)  we  have 
—,  =  ^~m  K^2  ™A[X'  (x2  +  Zy'2)  -  2a2x '] (48) 

62  z  d2z       (1  —K2)qg 

ei  =  62  =  -  id         =  -  id         =  [2a2-3(*'2+2/2](49) 

Oar  dy*       24Ejd2A 

d2  z         (l—K2)qgx'y' 

and       ,      ,  =  -  — —       - , (50) 

dx   dy  4  E  i  j  d  2 A 

These  expressions  satisfy  (20)  as  they  should,  for  (20)  is  inde- 
pendent of  the  directions  of  the  rectangular  axes  x  and  y. 

From  (49)  it  appears  that  ev  =  0  =  /s,  on  the  circumference  of 
the  circle  x'2jry'2  =  fa2,  which  passes  thru  the  points  where  the 
lines  of  contra-flexure  intersect. 

By  (19),  which  holds  for  any  rectangular  axes,  and  by  (50), 
we  find 

n'  =  1(1— K)  qx'  y'.  (26)' 

From  (26) '  it  appears  that  in  sections  by  all  vertical  planes 
parallel  to  the  diagonals,  the  twisting  increases  uniformly  with  the 
distance  from  the  diagonal. 

Hence  by  (9)  we  have 

5n' 

— SX 


j      V*-7/     »»  vy   j-j-dj  ' 

/6m; 
n  "  \8x' 

-        (^ 

2   W 


. . (28) 


It  thus  appears  that  the  same  law  holds  for  vertical  shearing 
stresses  on  planes  parallel  to  the  diagonals,  as  holds  in  (28)  for  planes 
parallel  to  the  edges  of  the  panel. 

In  standard  mushroom  designs  the  edges  of  the  diagonal  belts 
intersect  on  or  very  near  to  the  edges  of  the  side  belts.  That  makes 
the  middle  half  of  the  central  square  to  be  covered  by  double  belting, 
and  the  remainder  of  it  by  single  belting,  so  that  2 A  =  1.5^4.2  or 
perhaps  1.6  A2,  and  the  mean  value  of  A,  the  reinforcement  per 
unit  of  width  of  slab  here,  is  to  be  found  by  dividing  this  by  the 
width  of  a  belt,  which  is  7/8  a.  We  should  then  find  A  =  1.5  A2/ 
7/8  a  =  1.7  A2/ a.  But  this  mean  value  of  A  is  not  its  mean  effect- 
ive value  for  this  area,  because  the  reinforcement  is  so  disposed  as 


38  MEAN  REINFORCEMENT  STRESS  AT  PANEL  CENTER 

to  furnish  the  larger  values  of  /  in  the  central  diamond  just  where 
the  largest  true  applied  moments  and  stresses  occur.  The  mean 
value  of  A  in  the  central  diamond  is  2A2/7/Sa  =  2.3A2/a.  The 
mean  effective  value  lies  between  these  two  extremes,  probably 
nearer  the  latter  than  the  former.  A  similar  question  was  discussed 
in  connection  with  (40)  and  (41).  We  shall  assume  as  the  mean 
effective  reinforcement  in  this  central  rectangle, 

A  =  2A2/a,  and  /  =  2A2  ij  d\/a 
or  in  case  of  rectangular  panels 

I  =  4A2ijd22/(a  +  6) (51) 

In  case  of  rectangular  panels  the  term  2a2  in  (49)  should  be  replaced 
by  a2  +  b2  as  a  mean  value  to  make  it  depend  the  dimensions  of  the 
panel  symmetrically,  as  it  must.  Making  these  substitutions  in 
(49)  we  have  at  x  =  0  =  y  the  center  of  the  panel. 

W  (Li  +  L2)  (Lf  +  Ll)         C,W 
/s  =  Ee    = 


1024  L!  L2  A2  j  d2  256  A2  j  d2 

W(L,  +  La)  (L\  +  Ll)      Ci  W  L, 


(52) 


MI  =  2A2  j  d2  /s  = 

512  L!  L2  128 

where  Cl  =  l(Ll/L2  +  !)(!+  L22/Li2).     Take  j=0.89. 

If  1  >  L2  /L!  >  0.75  then  1  <  cx  <  1.042,  hence  Ci  varies  less  than  5% 
while  L2/Li  varies  by  25%  between  its  extreme  permissible  values. 
Ci  may  ordinarily  be  taken  as  unity,  or  may  be  found  with  sufficient 
precision  by  interpolation  between  the  values  just  given. 

The  steps  by  which  these  equations  (52)  were  deduced  may  not 
seem  conclusive,  since  they  are  not  rigorous.  They  need  be  only 
good,  working  approximations  for  the  purpose  for  which  they  will  be 
here  used,  viz,  to  show  that  the  stresses  at  the  center  of  the  panel 
are  less  than  those  at  the  mid  span  of  the  side  belts  in  case  AI  =  A2. 

The  value  of  d2  in  (52)  is  less  than  d1  in  (34),  but  always  more 
than  90%  of  it.  We  may  define  d2  as  the  vertical  distance  from  the 
center  of  the  second  and  upper  of  the  two  diagonal  belts  to  the  top  sur- 
face of  the  concrete.  We  may  assume  d2  =  0.9^  and  j  =  0.89  in 
(52),  and  then  we  may  compare  these  stresses  for  a  square  panel  as 
follows : — 

175 

•-« (53) 


where  /s'  refers  to  the  center  of  the  panel.      Even  were  the  smaller 


LINE    OF   ULTIMATE    WEAKNESS  39 

value  for  the  mean  reinforcement,  1.7  A2/a,  used  in  deriving  (52) 
and  (53),  the  stress  given  by  these  equations  would  not  exceed  that 
given  by  (34).  The  compressive  stress  /c  in  the  concrete  at  the 
center  of  the  panel  may  readily  reach  a  dangerous  value  in  case  the 
forms  are  removed  too  soon.  It  should  therefore  be  carefully  con- 
sidered in  each  case.  Here,  we  have  an  approximate  value  of  i  =  2/3 
and  (38)  then  becomes /c  =  /s/30  with  no  possible  assistance  from 
steel  reinforcement  since  that  is  all  on  the  bottom  of  the  slab.  An 
estimate  that  the  elastic  stress  in  the  steel  at  the  center  of  the  panel 
does  not  much  exceed  80%  of  that  at  the  middle  of  the  side  belt 
cannot  be  far  from  the  truth. 

While  this  is  undoubtedly  the  fact,  it  will  appear  on  further 
consideration  that  local  stresses  and  strains  which  exist  at  incipient 
failure  are  of  such  magnitude  as  to  make  the  weakest  points  of  the 
diagonal  belts  to  lie  ultimately  not  at  the  center,  but,  instead,  just 
outside  the  diamond  where  they  cross  each  other. 

Take  the  standard  case  where  the  central  diamond  reaches  just 
across  to  the  side  belts.  For  square  panels  imagine  a  circle  to  be 
drawn  concentric  with  each  column  of  radius  L/2.  Any  circle  at  a 
column  will  be  tangent  to  the  edges  of  four  diagonal  belts  across  the 
tops  of  the  four  columns  adjacent  to  it,  and  then  the  octagon  cir- 
cumscribing it,  whose  sides  cut  at  right  angles  all  the  belts  that  cross 
this  column  head,  intersects  but  a  single  belt  of  rods  as  every  point 
of  its  perimeter.  It  is  evident  that,  so  far  as  reinforcement  is  con- 
cerned, such  a  line  or  section  cuts  less  steel  per  unit  of  perimeter 
than  any  other  regular  figure  concentric  with  the  column  and  that 
the  reinforcement  is  entirely  symmetrically  disposed  about  the 
column  center,  so  that  in  case  of  equal  diagonal  and  side  belts,  it 
would  be  impossible  from  their  geometry  to  distinguish  the  one  from 
the  other  by  anything  inside  the  octagon.  That  fact  would  make  it 
inherently  probably  that  the  stresses  and  strains  of  the  rods  where 
they  cross  any  one  side  of  this  octagon  should  be  approximately  the 
same  ultimately  as  in  those  that  cross  any  other  side,  whether  they 
be  rods  in  a  diagonal  belt  or  in  a  side  belt.  And  what  will  be  at- 
tempted to  be  shown  immediately  is  that  ultimately  the  stresses 
and  strains  in  these  several  belts  approach  equality.  If  that  should 
be  established,  it  will  follow  from  the  conclusion  already  reached  as 
to  the  excess  of  the  stresses  and  strains  of  the  side  belt  over  those  at 
the  center  of  the  panel,  that  ultimately  those  at  the  edges  of  the 
ocatgon  exceed  those  in  the  same  rods  at  the  center  of  the  panel. 

The  qualification  implied  above  in  affirming  that  this  is  what 
will  occur  ultimately,  is  for  the  purpose  of  conveying  the  idea  that 


40  LOCAL  STRESSES  AND  STRAINS 

this  is  the  approximate  distribution  of  stresses  and  strains  which 
will  take  place  when  the  slab  is  sufficiently  loaded  to  bring  the  steel 
at  the  middle  of  the  side  belt  to  the  yield  point.  At  less  stress  than 
this  there  is  so  much  lag  in  the  distribution  of  the  effect  of  loading 
that  it  penetrates  to  the  various  parts  of  the  slab  unequally. 

Taking  up  now  the  deferred  proof  that  the  diagonal  rods  where 
they  cross  the  edge  of  the  octagon  are  subject  ultimately  to  the  same 
local  stresses  and  strains  as  the  direct  rods  of  the  side  belts;  note 
that  these  diagonal  rods  lie  in  a  triangular  area  between  two  side 
belts,  which  latter  experience  equal  elongations  e\  in  directions  at 
right  angles  to  each  other.  The  edges  of  the  triangle  in  which  the 
single  layer  of  diagonal  rods  lie  are  continuous  with  the  side  belts 
and  necessarily  experience  the  same  elongations,  which  are  propo- 
gated  from  the  side  belts  into  the  triangle  by  the  agency  of  horizontal 
shears  on  its  edges.  Such  equal  elongations  at  right  angles  imply 
the  same  elongation  in  every  direction  in  the  triangle,  as  appears 
from  the  fundamental  properties  of  equal  principal  stresses  and 
strains.  Hence  we  have  the  same  elongations  along  the  diagonal 
rods  as  along  the  rods  of  the  side  belts  at  the  edges.  The  existence 
of  an  ultimate  stress  and  strain  in  the  diagonal  belt  equal  to  that  in 
the  side  belt  would  require  that  the  cross  sections  A2  and  AI  of  the 
two  belts  should  be  equal,  altho  so  far  as  the  elastic  value  of  fs  at 
the  center  of  the  panel  is  concerned  A2  might  be  less  than  AI,  as  has 
been  already  shown  in  (52)  and  (53).  The  relationships  of  stress, 
load,  etc.,  for  this  ultimate  condition,  have  been  already  given  in  (37). 

Besides  the  stresses  and  strains  in  the  diagonal  belts,  just  in- 
vestigated, those  due  to  the  local  stretching  (arising  from  the  deflec- 
tions themselves)  exert  their  greatest  effect  on  the  rods  of  the  diagonal 
and  side  belts  just  in  the  region  of  the  line  of  weakest  section,  and 
partly  because  of  that  fact.  While  these  local  stresses  may  not  exceed 
10%  in  addition  to  those  already  present,  their  existence  should 
prevent  any  thought  of  taking  2 A  larger  than  AI  in  (37)  when  deriv- 
ing the  ultimate  stresses  at  the  yield  point.  Similar  results  may  be 
formulated  to  cover  cases  where  g  is  greater  or  less  than  7/16  L. 

It  is  perhaps  desirable  at  this  point  to  consider  a  little  more  at 
length  the  matter  of  local  stretching  in  a  slab.  It  is  impossible  for  a 
continuous  flat  floor  slab  to  undergo  the  deflections  which  we  are 
treating,  consisting  of  convexities,  concavities,  etc.,  without  local 
stretching  to  allow  this  to  occur.  A  floor  slab  of  many  panels  does 
not  undergo  any  change  of  its  total  linear  dimensions  which  would 
account  for  these  corrugations.  A  continuous  beam  under  flexure 
would  have  its  extremities  drawn  toward  each  other.  But  not  so 


ACTUAL   DEFLECTIONS    IN    SIDE    BELTS  41 

to  any  such  extent  with  a  slab.  Such  contractions  are  resisted  by 
local  circumferential  strains  which  result  in  true  stresses.  An 
investigation  of  such  stresses  leads  to  the  conclusion  just  stated  that 
in  general  they  cannot  exceed  10%  of  the  ordinary  stresses  due  to 
slab  bending  when  they  are  left  out  of  the  consideration.  For  this 
reason  a  single  panel  alone  will  not  function  precisely  in  the  same 
way  as  a  panel  in  a  floor  of  many  panels. 

12.  Actual  deflections  are  distances  which  any  given  points 
of  a  slab  sink  down  by  reason  of  the  application  of  a  given  load,  and 
their  theoretical  values  are  to  be  computed  by  help  of  the  formulas 
which  have  been  developed  for  z  in  the  various  areas  into  which  the 
panel  has  been  divided. 

We  shall  now  make  a  slight  modification  in  our  definition  of  the 
level  of  the  origin  of  coordinates,  and  shall  take  it  at  the  upper  or 
lower  plane  surface  of  the  flat  slab  before  flexure,  in  which  surface 
the  axes  of  x  and  y  are  assumed  to  lie.  It  is  of  no  consequence 
whether  it  be  the  upper  or  the  lower  surface  which  is  assumed,  the 
equations  will  be  the  same  in  either  case.  The  reason  for  this  new 
definition  of  the  position  of  the  origin  is  this:  Each  kind  of  partial 
area  into  which  the  slab  has  been  supposed  to  be  subdivided  has  its 
neutral  surface  at  a  different  depth  in  the  slab,  and  so  it  does  not 
furnish  a  single  suitable  level  from  which  to  reckon  deflections,  as 
does  the  upper  or  lower  surface  of  the  slab.  None  of  the  equations 
which  have  been  derived  in  this  paper  will  undergo  any  modification 
by  reason  of  this  change  of  definition.  It  has  been  assumed  that 
each  kind  of  area  has  a  separate  value  of  /  which  remains  constant 
thruout,  so  that  the  neutral  surfaces  of  different  areas  do  not  join 
at  their  edges.  As  previously  explained  this  is  of  no  consequence 
mechanically  by  reason  of  the  zero  true  moments  that  exist  at  these 
edges.  The  modification  just  introduced  avoids  the  geometrical 
perplexities  arising  from  this  discontinuity  of  neutral  surfaces. 

Deflections  in  the  side  belt  area  between  the  lines  of  contra- 
flexure  (24)  are  to  be  found  from  (30),  or  (31),  and  (32).  To  find 
the  deflection  or  difference  of  level  in  the  mid  side  belt  between 
x  =  0,  y  =  6,  and  x  =  %  a  V^3,  y  =  b,  substitute  these  values  in  (30), 
take  i  =  0.71,  j  =  0.91,  k  =  0.5  and  subtract  the  value  z  at  the 
second  point  from  that  at  the  first  point,  which  gives  the  following 
value  of  the  deflection  of  the  one  point  below  the  other: 

WL\ 


42  ACTUAL  DEFLECTIONS  IN  CENTRAL  AREA,  IN  HEAD 

in  which  hi  is  the  vertical  distance  from  the  center  of  the  single  belt 
of  rods  at  the  mid  span  of  the  side  belt  to  the  effective  top  of  the 
slab,  considering  the  strip  fill  or  other  concrete  finish  at  its  effective 
value. 

In  the  same  manner  take  the  difference  of  level  in  the  central 

rectangle  bounded  by  the  lines  of  contraflexure  between  the  center 

point  at  x  =  0,  y  =  0  and  the  corner  x  =  f  a  Vs,  y  =  J  b  V^3  by  using  (21) 

and  (51)  and  introducing  the  values  i  =  2/3,     j  =  0.89,  etc.,  and 

C2  =  l/4(Li/L2  +!)(!+  L\/L\},  then: 

C2W  L\ 
A  z2  =  -  —  —  -  --  .........................  (55) 

6.56  x  1010  d\  A2 

in  which  A2  is  the  cross  section  of  one  diagonal  belt  and  h2  is  the 
vertical  distance  from  the  center  of  the  upper  or  second  diagonal 
belt  to  the  effective  upper  surface  of  the  panel  at  its  center. 

On  evaluating  C2  above,  we  find 
when        l>L2/Li  >  0.75 
then         1>      C2       >0.77 

hence  we  may  with  sufficient  accuracy  for  practical  purposes  assume 
C2  =  L2/L,  ...................................  (56) 


Deflections  in  the  mushroom  area  between  lines  of  contraflexure 
(24)  are  to  be  derived  from  (39)  (40)  and  (41)  by  introducing  i  =  J, 
j  =  0.83,  k  =  0.5  and  SA  =  7.5  A\.  Assuming  the  diameter  of 
the  cap  to  be  0.2Li  we  have,  at  its  edge  where  x  =  0.8a,  y  =  b,  from 
(39) 

W  Ll  (Li/La  +1)  /36y 

19.1  xl010dl  Al  VlOO/ 

The  value  of  z  at  the  edge  of  the  mushroom  area,  where  x  =  J  a  V%, 
y  =  6,  is  to  be  obtained  from  (57)  by  replacing  the  last  factor  by 
4/9;  and  the  deflection  between  the  edge  of  the  cap  and  the  edge  of 
the  mushroom  obtained  by  taking  the  difference  of  these  quantities 
is  as  follows: 

WL3l(Ll/L2  +  l) 

A  z3  =  -  ~ 


in  which  A3  is  the  vertical  distance  of  the  center  of  the  third  layer  of 
reinforcing  rods  over  the  edge  of  the  cap  above  the  bottom  surface 
of  the  slab. 


TOTAL  DEFLECTIONS  BELOW  EDGE  OF  CAP  43 

Similar  expressions  may  be  obtained  for  the  values  of  z  and  Az 
on  the  side  parallel  to  y,  where  x  =  a  at  y  =  0.86,  and  y  =  \  bV  3, 
by  exchanging  LI  and  L2  in  (57)  and  (58). 

Take  half  the  sum  of  (57)  and  the  corresponding  values  so  ob- 
tained at  x  =  a,  y  =  0.86,  as  the  value  of  z  at  the  edge  of  the  cap 
where  it  is  intersected  by  the  diagonal  of  the  panel,  viz. 

W  (Li  +  L2)  (L\  +  LJ)  /  36  y 
38.2  x  1010  LI  L2  d\  Al  \100/ 

and  subtract  this  from  the  value  of  z  on  the  diagonal  at  the  corner 
of  the  mushroom  area  where  x  =  J  aV%  y  =  J  6V^3  and  we  have 

C2W  L\ 

A  z4  =  -  — - — (60) 

12.5  x  10     d%  AI 

as  the  deflection  along  the  diagonal  between  the  edge  of  the  cap  and 
the  intersection  of  the  lines  of  contraflexure,  in  which  C2  and  h3  are 
as  previously  defined. 

Let      DI  =  Azi  +  Az3\ 

and  D2  =  Az2  +  Az4J 
in  which  L>i  is  the  deflection  of  the  mid  point  of  the  side  belt  below 
the  edge  of  the  cap,  and  D2  is  the  deflection  of  center  of  the  panel 
below  the  edge  of  the  cap. 

The  proportionate  deflections  of  these  points  are  obtained  by 
dividing  by  the  spans,  viz:  Dl/Li  and  D2/  V L\  +  Lf. 

13.  Estimated  proportionate  deflections  may  be  obtained  from 
(61)  under  such  circumstances  as  to  convey  reliable  information 
respecting  what  may  be  reasonably  expected.  Let  h  =  the  total 
thickness  of  the  slab.  The  limiting  values  of  the  thickness  of 
standard  mushroom  construction  are  expressed  as  follows: 

Li/20>A>Li/35, (62) 

and  assuming  that  the  reinforcing  rods  are  1/2"  rounds  with  1/2" 
covering  of  concrete  we  shall  have  from  the  definitions  of  diy  d2  and 
d3,  already  given 

h  =  di  +  0.75  =  d2  +  1.25  =  d3  +  1.75 (63) 

Substituting  these  in  (62)  etc.  we  have 

Li/20  —  0.75  >  ^  >  Li/35  —  0.75   ] 

Li/20  —  1.25  >  d2  >  Li/35  —  1.25    }• (64) 

Li/20  —  1.75  >  d3  >  Li/35  —  1.75 


44 


PROPORTIONATE  DEFLECTIONS 


If  it  be  assumed  that  we  are  dealing  with  medium  sized  panels 
about  20'  x20'  (64),  may  be  written  in  the  form:  — 

(1  —  0.062)  Li/20  >  di  >  (1  —  0.02)  Li/35 
(1  —  0.1)  Lj/20    >d2  >  (1  —  0.036)  Lj/35 
(1  —  0.15)  L!/20  >  d3>  (I—  0.05)  Lx/35 


or, 


0.94 


0.98 


20        L!          35 

0.90      d2        0.964 

>  —  > 

20        Li          35 


0.85 
— 
20 


0.95 

-- 

35 


(65) 


In  (54),  (55),  (58)  and  (60)  replace  W  Lv  by  its  value  given  in  (34), 
viz,    175  di  A i  fs)  and  we  have 


Llfn 

A« 

6.11  x  108  dl 

Z2    — 

3.75  x  108  dl  A2 

A9 

di  LI  (L/i  /  L/2  H-  1)  fs 

%3    — 

A;y 

34.3  x  108  dl 

n    j    jl  f 
L2  di  LI  Js 

#4  — 

7.14  x  108  dl 

(66) 


in  which  fs  is  the  greatest  stress  in  the  steel,  i.  e.,  at  the  mid  side 
belt,  employed  here  to  express  deflections  instead  of  expressing  them 
in  terms  of  panel  load  as  was  done  previously. 


RELATIVE    DEFLECTIONS    AT   MID    SPAN 


45 


Introduce  into  (66)  the  numerical  values  given  in  (65)  which 
will  then  express  limiting  values  of  deflection  for  medium  spans. 
For  simplicity  let  LI  =  L2  then: 


287  > 


162   > 


105Azi 


>    170 


>    106 


105Az2 
660  >        1         >    451 


275   > 


105Az4 


>    188 


(67) 


By  (61)  we  have  the  proportionate  deflection  of  the  side  and  diagonal 
belts  as  follows: — 


f  i       1 1  fs      A     r  i       in 

i        -  +  -  ^r<-       <          -  +  - 

1.287       660J  105        LI        Ll70       45lJ 


105 


-  +  —  -A-^A^LL  +±"  ^ 

Ll62   275 J  105  V2   Lx  V2   Ll06   188 J  105  V2 


/s 


200  x  105 


< 


< 


-  < 


123. 4  x  105 

/s 


141.4xl05      L!  V2        95.9x10 


1  D2  I 

If  fa  =   16000,  <  :   < 

884        L!  V2        600 


If  fs  =  24000, 

590 


D2  1 

< 


/i  V2        400 


1  £>2  1 

If  /s  =  32000,  <   -  < 

442         L!  V2         300 

Larger  spans  then  20 ',  or  smaller  steel  than  1/2"  round,  or  L2<Li 
will  reduce  the   above  values  somewhat,   while  smaller  spans   or 


(68) 


(69) 


46  THEORETICAL    AND    EXPERIMENTAL   DEFLECTIONS    COMPARED 

larger  steel  will  increase  these  values,  all  of  which  can  in  each  case 
be  submitted  to  calculation  by  the  methods  here  developed. 

To  recur  at  this  point  to  the  expression  for  the  deflection  D2  in 
terms  of  the  panel  load  W  by  help  of  (55),  (60)  and  (61) 

C2W  L] 


^r JL i    j 

.1  |_6.56d|       12.5  d\\ 


By  (65)  we  find 

90       d*       96.4 


85       d3         95 

and  using  this  inequality  to  eliminate  d3  from  (70)  we  find  after 
reduction 

C2  W  L\  C2W  L\ 

D 


4.46  x  1010  dl  Al  4.33  x  1010  d\  Al 

from  which  we  may  write  as  a  mean  value 
C2W  L\ 


" 


4.4x10 


(71) 


The  empirical  deflection  formula  given  on  page  29  of  Turner's  Con- 
crete Steel  Construction,  when  written  in  these  units,  is 

WL\ 
D2  =  4.84xl010dUr 

This  is  identical  with  (71)  when  C2  =  0.909,  and  diverges  from  it 
slightly  for  other  admissible  values  of  C2.  The  practical  agreement 
of  (71)  and  (72)  affords  a  second  confirmation  of  the  theoretical 
deductions  made  thus  far,  and  this  taken  in  conjunction  with  the 
practical  identity  of  formulas  (34)  and  (35),  the  theoretical  and 
empirical  expressions  for  the  maximum  tensile  stresses  in  the  rein- 
forcement, furnishes  what  on  the  theory  of  probabilities  may  be 
regarded  as  so  strong  a  probability  of  the  general  trustworthiness 
of  the,  entire  theory  as  to  exclude  any  rational  suppositition  to  the 
contrary. 

The  various  formulas  for  stresses  and  for  deflections  which  have 
been  developed  in  this  paper  have  been  obtained  under  the  express 
proviso  that  the  panel  under  consideration  was  assumed  to  be  one 
of  a  practically  unlimited  number  of  equal  panels  constituting  a 
continuous  slab,  all  of  which  are  loaded  uniformly  and  equally.  The 


MUSHROOM    PANELS    INDEPENDENT  47 

question  at  once  arises  as  to  the  amount  and  kind  of  deviations  from 
these  formulas  which  will  occur  by  reason  either  of  discontinuity  of 
slab  or  loading,  such  as  occurs  at  the  outside  panels  of  a  slab  or  at 
panels  surrounded  partly  or  entirely  by  others  not  loaded.  The 
answer  to  this  question  depends  very  largely  upon  the  construction 
of  the  flat  slab  itself. 

In  the  standard  mushroom  construction  it  has  been  found  that 
the  stresses  and  deflections  of  any  panel  are  almost  entirely  inde- 
pendent of  those  in  surrounding  panels.  This  is  due  to  the  fact 
that  the  mushroom  head  is  an  integral  part  of  the  supporting  column 
in  such  a  manner  that  it  is  impossible  for  it  to  tilt  appreciably  over 
the  column  under  the  action  of  any  eccentric  or  unequal  loading  of 
panels  near  it.  When  single  panels  have  been  loaded  with  test 
loads,  no  appreciable  deflections  have  been  discoverable  in  sur- 
rounding panels,  and  no  greater  stresses  and  deflections  have  been 
discovered  than  were  to  be  expected  in  case  surrounding  panels  were 
loaded  also.  Future  careful  investigation  of  this  may  reveal 
measureable  effects  of  this  kind,  but  they  must  be  small. 

A  like  statement  cannot  be  made  of  other  systems  of  flat  slab 
construction  where  the  reinforcement  over  the  top  of  the  column  is 
not  an  integral  part  of  the  column  reinforcement  itself.  Tests  on 
these  systems  have  shown  clearly  the  effects  of  the  tipping  of  the 
part  of  the  slab  on  the  top  of  the  column,  and  lack  of  stiffness  of 
head,  in  the  increase  of  the  deflection  of  the  single  loaded  panel  over 
the  deflection  to  be  expected  in  case  of  multiple  loaded  panels,  and 
especially  in  the  disturbance  of  the  equality  of  the  stress  in  the  other- 
wise equal  stresses  in  the  rods  of  the  side  belts.  Such  distrubance, 
by  increasing  the  stress  in  part  of  these  rods,  would  necessitate  larger 
reinforcement  in  the  side  belts  of  such  systems  than  would  be  re- 
quired in  mushroom  slabs.  The  great  stiffness  of  the  mushroom 
head  is  also  of  prime  importance  in  taking  care  of  accidental  and 
unusual  strains  liable  to  occur  in  the  removal  of  forms  from  under 
insufficiently  cured  slabs. 

14.  In  considering  the  design  of  the  ring  rods  and  radial 
cantilever  rods  of  the  mushroom  head,  it  should  be  borne  in  mind 
that  they  occupy  a  position  in  such  close  proximity  to  the  level 
of  the  neutral  surface  as  to  prevent  them  from  being  subjected 
to  severe  tensile  or  compressive  stresses  by  reason  of  the  bending 
of  the  slab  as  a  whole.  Their  principal  function  as  slab  mem- 
bers is  to  resist  shearing  stresses  and  the  bending  stresses  due  to 


48  VERTICAL  SHEAR  AROUND  COLUMN  CAPS 

local  bending.  Their  total  longitudinal  stresses  are  too  small  in 
comparison  to  require  consideration. 

Let  a  cylindrical  surface  be  imagined  to  be  drawn  concentric 
with  a  column  to  intersect  the  slab,  then  the  total  vertical  shearing 
stress  which  is  distributed  on  the  surface  of  intersection  is  equal  to 
the  total  panel  load  W  diminished  by  the  amount  of  that  part  of  the 
panel  load  lying  inside  the  cylinder.  If  the  cylinder  be  not  large, 
the  total  shear  may  be  taken  as  approximately  equal  to  W  itself. 

It  is  evident  that  the  smaller  the  diameter  may  be  that  is 
assumed  for  this  cylinder,  the  greater  will  be  the  intensity  of  the 
vertical  shear  on  its  surface  and  that  for  two  reasons :  First,  because 
the  totsl  load  thus  carried  to  the  column  will  be  greater  the  smaller 
the  diameter,  and  second  because  the  surface  over  which  the  total 
shear  will  be  distributed  decreases  with  its  diameter. 

The  result  of  this  is  that  the  dangerous  section  for  shear  is  the 
cylindrical  surface  at  the  edge  of  the  cap.  For  cylinders  smaller 
than  this  the  increased  vertical  thickness  of  the  cap  diminishes  the 
intensity  of  the  shear.  We  proceed  therefore  to  consider  the  manner 
in  which  the  total  vertical  shearing  stress  of  approximately  W  in 
amount  is  distributed  in  the  material  of  the  cylindrical  surface  at  the 
edge  of  the  cap. 

In  a  beam  or  slab  the  horizontal  shearing  stresses  due  to  bending 
reach  a  maximum  at  the  neutral  surface.  It  is  a  fundamental  con- 
dition of  equilibrium  that  shearing  stresses  on  planes  at  right  angles 
shall  be  equal,  and  it  is  this  condition  that  determines  the  distribu- 
tion of  the  vertical  shears,  which  are  at  right  angles  to  the  horizontal 
shears  resulting  from  bending  the  slab  as  a  whole.  From  this  we 
have  the  well  known  fact  that  the  vertical  shear  varies  from  zero 
at  the  upper  and  lower  surfaces  to  a  maximum  at  the  neutral  surface, 
and  this  is  necessarily  the  manner  in  which  the  total  shear  is  dis- 
tributed at  the  edge  of  the  cap.  The  top  belt  of  rods  will  be  sub- 
jected to  comparatively  small  shearing  stresses,  and  successive 
layers  of  rods  will  be  under  larger  and  larger  shearing  stresses  by 
reason  of  their  greater  nearness  to  the  neutral  surface,  while  the 
total  shear  borne  by  the  radial  rods  near  the  neutral  surface  will  be 
much  larger  than  that  upon  the  others.  The  shearing  stress  in 
the  concrete  will  need  to  be  considered  also. 

It  is  to  be  noticed  that  all  the  steel  of  the  belts  and  mushroom 
head  act  together  without  the  necessity  of  supposing  large  com- 
pressive  stresses  in  the  concrete  to  transmit  vertical  forces,  because 
the  belts  of  reinforcement  rest  directly  upon  each  other,  and  these 
in  turn  upon  the  ring  rods  and  radial  rods,  all  in  metallic  contact 


VERTICAL  SHEAR  IN  RADIAL  RODS,  ETC.  49 

with  each  other,  in  the  mushroom  head,  and  so  they  transmit  and 
adjust  the  distribution  of  stresses  within  the  system  to  a  very  large 
extent  independently  of  the  concrete. 

We  can  then  safely  assign  moderate  values  of  the  shearing 
stress  to  each  of  the  elements  that  constitute  the  slab  at  the  edge 
of  the  cap,  with  the  assurance  that  they  will  each  play  a  part  in 
general  accordance  with  the  distribution  which  has  been  already 
explained. 

The  mushroom  is  constructed  of  great  strength  and  stiffness 
not  merely  to  effect  the  results  which  have  appeared  previously  in 
the  course  of  the  investigation  but  also  to  ensure  the  stability  of  the 
slab  in  case  of  unexpected  or  accidental  stresses  due  to  the  too  early 
removal  of  the  forms,  before  the  slab  is  well  cured,  at  a  time  when 
the  only  load  to  which  it  is  subjected  is  due  to  the  weight  of  the 
structure  itself. 

The  working  load  to  be  assumed  in  designing  the  mushroom 
may  be  taken  as  the  dead  load  of  a  single  slab  plus  the  design  load, 
provided  sufficiently  low  values  of  the  shearing  stresses  be  assumed 
in  the  cross  sections  of  steel  and  concrete  at  the  edge  of  the  cap 
for  the  support  of  this  working  load,  as  follows : 

For  slabs  having  a  thickness  of  h  =  L/35  a  mean  working 
shearing  stress  of  2000  Ibs.  per  square  inch  at  the  right  cross  section 
of  each  reinforcing  rod  which  crosses  the  edge  of  the  cap,  a  mean 
shearing  stress  of  40  Ibs.  per  square  inch  in  the  vertical  cylindrical 
section  of  the  concrete  at  the  edge  of  the  cap,  and  8000  Ibs.  per 
square  inch  of  right  cross  section  of  each  radial  rod. 

For  slabs  having  a  thickness  of  h  =  L/20  the  intensities  just 
given  may  be  safely  increased  by  50  per  cent  for  reasons  that  will 
be  explained  later.  For  slabs  of  intermediate  thickness  increase 
the  intensities  proportionately. 

These  values  are  sufficiently  low  to  enable  the  structure  to  sup- 
port itself  before  the  concrete  is  very  thoroughly  cured,  and  the 
head  so  designed  will  be  found  after  it  is  well  cured  to  be  so  pro- 
portioned as  to  carry  safely  a  test  load  of  double  the  live  and  dead 
loads  for  which  it  was  designed. 

In  this  connection  it  seems  desirable  to  investigate  what  takes 
place  in  case  of  overloading  and  incipient  failure  of  an  insufficiently 
cured  slab,  or  one  unduly  weakened  by  thawing  of  partially  frozen 
concrete.  Suppose  that  under  such  circumstances  a  shearing  crack 
were  formed  extending  completely  thru  the  head  at  the  edge  of  the 
cap,  and  we  wish  to  investigate  the  stresses  and  behavior  of  the  rods 


50  STRESSES    IN    RADIAL   RODS 

that  cross  the  crack  at  which  shearing  deformation  has  begun  to  take 
place.  Designate  the  position  of  the  crack  by  X. 

The  total  vertical  shearing  stress  on  a  radial  rod  at  X  is  the 
sum  of  two  parts  found  as  follows:  First,  the  vertical  reaction  at 
the  top  of  a  column  is  made  up  of  the  vertical  reaction  of  the  con- 
crete core  of  the  column  and  the  reactions  of  its  vertical  reinforcing 
rods.  Call  the  vertical  reaction  of  one  of  these  rods  V\.  The  rod 
is  bent  over  radially  and  Vi  expresses  also  the  amount  of  the  vertical 
shear  in  that  rod  where  it  starts  out  radially  from  the  column. 
Between  this  point  and  X  for  a  distance  which  measures  usually 
from  9  to  12  inches,  the  rod  experiences  the  supporting  pressure  of 
the  concrete  in  the  cap  under  it  to  a  total  amount  which  we  will 
designate  by  V2.  The  total  shear  in  the  radial  rod  at  X  will  then 
amount  to 

V  =  V,+  V2 (73) 

provided  we  neglect  the  weight  of  that  small  part  of  the  actual  load 
of  the  slab  which  lies  directly  over  this  piece  of  the  rod  and  may 
be  regarded  as  resting  upon  it.  This  portion  of  the  radial  rod  of 
length  I  is  a  cantilever  fixed  at  one  end  in  the  top  of  the  column,  and 
carrying  a  load  V  at  the  other  end  with  a  supporting  pressure  under- 
neath of  total  amount  V2  whose  intensity  is  greatest  at  X  and  gradu- 
ually  decreases  along  /  from  X  to  the  fixed  end.  The  rod  has  a 
point  of  contraflexure  and  zero  moment  at  X.  The  portion  of  the 
rod  outside  the  crack  has  a  fixed  point  in  the  slab  at  the  place  where 
it  supports  the  inner  ring  rod,  at  a  distance  from  X  which  should 
not  exceed  I  as  just  defined.  Similar  conditions  hold  for  this  length; 
i.  e.  there  will  be  a  totol  shear  in  the  radial  rod  at  a  point  just  inside 
the  inner  ring,  rod  due  to  its  total  shear  outside  this  ring  rod  and  to 
the  vertical  loading  imparted  to  it  by  the  ring  rod  itself.  To  this 
must  be  added  the  downward  pressure  of  the  concrete  between  the 
inner  ring  rod  and  X.  All  these,  together,  constitute  the  total 
shear  — V  at  X,  in  equilibrium  with  the  reaction  -f  V  already  ob- 
tained at  that  point. 

We  shall  discuss  separately  the  action  of  V\  and  V2  upon  a  radial 
rod.  A  load  V\  at  the  end  of  a  cantilever  of  length  I  causes  a 

deflection  of  amount  zl  =  \Vi  f  /El (74) 

in  which  1=  TT  £4/64  where  Z  =  the  thickness  of  the  rod. 

Also  V1  =  s1  A     ,  A=  7r*2/4 

in  which  Si  =  the  mean  shearing  stress  per  square  unit  of  cross  sec- 
tion and  A  is  the  cross  section  of  the  rod.  Hence 

(75) 


STRESSES  IN  RADIAL  RODS  51 

which  shows  that  so  far  as  V\  is  concerned,  for  any  given  displace- 
ment z\  the  shearing  stress  carried  per  square  unit  of  rod  will  be  pro- 
portional to  the  square  of  its  diameter,  and  up  to  its  permissible 
limiting  shearing  resistance,  each  unit  of  section  of  such  a  rod  will 
be  effective  in  proportion  to  the  square  of  its  diameter.  For  econ- 
omical construction,  this  will  require  the  radial  rods  to  be  few  and 
large,  rather  than  numerous  and  small.  The  bending  moment  is 
greatest  at  the  distance  I  from  X  and  amounts  to  V\  I.  The  stress 
in  the  extreme  fiber  due  to  the  bending  moment  V\  I  in  the  rod  is 
Pl  =  Vilt/2I  =  Ssil/t (76) 

This  equation  shows  that  the  stress  in  the  extreme  fiber  is  so  very 
large  at  the  fixed  end  of  the  rod  compared  with  the  shear  at  X  that 
so  far  as  Vi  is  concerned  the  rod  will  suffer  permanent  deformation 
by  bending  long  before  there  is  any  danger  of  its  shearing.  V\  is  so 
large  compared  with  V2  that  this  conclusion  will  not  be  altered  when 
we  come  to  consider  the  combined  action  of  V%. 

Incipient  failure  of  this  kind  will  therefore  cause  distortion  and 
sag  without  collapse.  In  case  such  sag  as  occurs  in  this  case  is 
detected  underneath  the  head  around  the  cap,  the  slab  should  be 
blocked  up  at  once  and  the  concrete  picked  out  at  all  parts  showing 
facture.  This  should  then  be  refilled  with  a  stronger  concrete 
which  will  set  rapidly.  Such  repair  should  not  weaken  the  slab. 

Whenever  the  intensity  with  which  a  radial  rod  presses  upon 
the  concrete  at  the  edge  of  a  crack  at  X  passes  the  compressive 
strength  /„  of  the  concrete,  it  must  begin  to  yield.  Afc  this  instant 
we  shall  have  a  pressure  of  the  concrete  against  the  rod  which  gradu- 
ally diminishes  as  we  pass  along  the  rod  from  X  to  the  distance  I, 
where  it  becomes  zero.  We  shall  assume  that  the  pressure  dimin- 
ishes uniformly  with  this  distance.  This  may  not  be  precisely  cor- 
rect, but  cannot  be  much  in  error.  If  the  shear  ¥2  at  X  is  the  sole 
cause  of  this  pressure,  then  F2  =  |  tlfcj  and  f  ¥2  I  =  $tffe 
is  the  bending  moment  in  the  rod  at  the  distance  I,  due  to  V2  at  X 
and  the  pressure  distributed  along  I. 

It  will  be  found  that  these  produce  a  deflection 

z2  =  3  /c  P/2QEI  =  0.3  f  V2/EI (77) 

a  unit  shear  of 

S2  =  V2/A  =  z2  E  *2/4.8  /3 (78) 

and  a  stress  on  the  extreme  fiber  at  a  distance  I  amounting  to 

p2  =  V2  1 1/31  =  16s2  l/t (79) 


52  SHEAR    IN    OUTER    RING    RODS 

It  thus  appears  that  the  equations  expressing  the  action  of  V2 
are  precisely  similar  to  those  for  Vi,  differing  only  in  their  numerical 
coefficients,  and  consequently  all  the  statements  as  to  the  resistance 
of  the  radial  rods  under  the  action  of  Vi  hold  for  the  action  of 
F!  and  V2  together  in  the  case  of  given  initial  deformations, 
Zi  =  z2  at  X. 

While  the  preceding  investigation  has,  in  order  to  make  ideas 
explicit,  ostensibly  assumed  a  crack  at  X  and  an  initial  small  shear- 
ing deformation  at  X,  the  investigation  applies  equally  well  to  the 
elastic  shearing  deformation  of  the  concrete  at  the  dangerous 
section  in  which  case  the  total  shearing  stress  will  consist  of  an  addi- 
tional componenent  due  to  the  resistance  of  the  concrete,  which 
however  may  for  additional  safety  be  neglected.  If  the  assumed 
deformation  be  confined  within  limits  so  small  that  the  concrete 
is  able  to  endure  it  without  cracking  then  the  preceding  investiga- 
tion may  properly  be  applied  to  it.  It  is  right  here  that  the  thick- 
ness of  the  radial  rods  is  able  to  render  its  most  effective  service, 
for  it  appears  from  (75)  and  (78)  that  any  permissible  intensity  of 
shear  may  be  developed  in  the  radial  rods  by  making  them  of  suit- 
able thickness,  even  tho  the  deflection  be  kept  within  the  elastic 
limits  of  the  concrete. 

As  already  stated  we  must  not  overlook  the  fact  that  the  major 
stresses  here  are  those  under  the  head  of  V\,  which  are  due  to  the 
direct  metallic  contacts  of  the  steel  rods  resting  one  upon 
another,  where  large  stresses  are  transmitted  and  pass  independ- 
ently of  the  concrete  except  for  the  distortions  of  the  steel  which 
meet  resistance,  and  the  secondary  reactions  such  as  have  been 
treated  in  a  single  aspect  while  investigating  the  action  of  F2. 

It  is  due  to  this  fact  that  large  shearing  stresses  may  be  safely 
borne  by  the  slab  at  and  near  the  edge  of  the  cap,  which  the  concrete 
mostly  escapes,  it  merely  furnishing  some  lateral  stiffening  to  the 
steel.  On  this  principle  the  outer  ring  rod  should  have  a  cross 
section  not  much  less  than  one  half  that  of  the  radial  rods  on  which 
it  rests.  For,  this  arrangement  provides  for  the  transferal  to  the 
radial  rods  of  all  the  shear  the  ring  rod  is  able  to  carry,  it  being  in 
double  shear  compared  with  the  radial  rod  it  rests  on. 

It  is  impossible  to  determine  the  cross  section  of  the  inner  ring 
rod,  with  the  same  defmiteness  as  that  of  the  radial  rods,  but 
that  is  unimportant.  Its  position  has  already  been  fixed  as  not 
more  than  I  from  the  edge  of  the  cap,  where  I  is  the  distance  from 
the  top  hoop  or  collar  band  of  the  column  to  the  edge  of  the  cap. 


STRESSES    IN    CONCRETE    OF    HEAD  53 

The  vertical  shearing  stresses  may  be  regarded  as  sufficiently 
resisted  outside  the  mushroom  by  the  concrete  alone.  The  critical 
cylindrical  surface  separating  those  areas  where  the  shear  may  be 
assumed  to  be  safely  carried  by  concrete  alone,  from  those  areas 
where  the  steel  may  be  relied  on  to  carry  as  much  of  the  shear  as 
may  be  required,  should  evidently  be  taken  somewhat  inside  the 
outer  ring  rod,  but  just  where  is  of  no  particular  consequence. 

The  supposition  of  the  existence  of  a  crack  at  X,  either  actual 
or  potential,  on  which  our  computation  of  the  stresses  in  the  radial 
rods  has  been  based,  is  sufficiently  satisfactory  so  far  as  the  rods 
themselves  are  concerned ;  but  it  seems  desirable  to  consider  in  more 
detail  the  phenomena  attending  the  development  of  the  stresses  in 
the  concrete  at  and  near  the  edge  of  the  cap,  especially  in  soft  con- 
crete when  the  limit  of  its  compressive  resistance  is  reached  in  this 
region. . 

The  horizontal  compressive  resistance  of  the  concrete  at  the 
lower  surface  of  the  slab  is  that  already  treated  in  (38),  and  it  is  our 
present  object  to  consider  how  that  is  to  be  combined  with  the 
vertical  supporting  pressures  under  the  radial  rods,  and  with  the 
horizontal  and  vertical  shears  in  the  slab  due  to  bending.  These 
latter  are  greatest  in  the  neutral  surface,  as  has  been  previously 
stated,  and  according  the  general  theory  of  stresses  are  equivalent 
to,  and  may  be  replaced  by,  a  compression  and  a  tension  in  the  mate- 
rial respectively  at  45°  with  the  vertical  (and  mutually  at  right 
angles)  of  the  same  intensity  as  the  shear.  It  is  evident  that  the 
combination  and  resultant  of  these  three  compressive  stresses 
would  form  the  dangerous  element  in  the  stress,  since  the  single 
tensile  element  would  be  relatively  unimportant,  and  it  would  find 
assistance  in  its  resistance  from  the  steel  running  in  a  direction  thru 
the  concrete  such  as  to  afford  it  substantial  support.  This  direction 
is  that  of  the  straight  lines  on  the  surface  of  a  right  cone  whose 
vertex  is  above  the  center  of  the  column  and  whose  slope  is  1  to  1 . 

Consider  now  two  of  the  elements  of  the  compression  in  the 
concrete  around  the  cap,  viz,  the  horizontal  compression  which  is  a 
maximum  at  the  lower  surface  and  zero  at  the  neutral  surface,  and 
that  due  to  shear  which  is  parallel  to  the  sides  of  a  right  cone  with 
vertex  downward,  whose  sides  have  an  upward  and  outward  slope 
of  1  to  1,  while  its  intensity  is  so  distributed  that  it  is  zero  at  the 
bottom  of  the  slab  and  greatest  at  the  neutral  surface.  It  appears 
consequently  that  the  lines  of  greatest  compression  in  the  concrete 
due  to  the  combination  of  these  two  elements  of  compression  would 


54  COMPRESSION  ON  CONCRETE  OF  HEAD 

lie  in  vertical  planes  on  a  bowl  or  saucer-shaped  surface  that  is  hori- 
zontal at  the  edge  of  the  cap  and  inclined  at  a  slope  of  45°  at  the 
neutral  surface ;  and  if  the  concrete  were  to  crush  under  these  stresses 
alone,  the  surface  of  fracture  would  have  the  shape  indicated  in- 
stead of  that  of  the  cylindrical  surface  previously  assumed.  This 
change  would  not,  however,  materially  affect  the  computations  we 
have  made  of  stresses  in  steel;  it  merely  serves  to  fix  more  definitely 
the  position  of  the  points  of  contra-flexure  of  the  radial  rods. 

But  there  is  still  one  further  element  or  component  of  the  total 
compression  in  the  concrete  to  be  considered  and  combined  with 
those  just  treated  in  order  to  arrive  at  the  resultant  or  total  com- 
pression. This  componenent  is  that  due  to  the  concentrated  press- 
ures underneath  each  of  the  radial  rods.  These  rods  are  at  some 
distance  apart  circumferentially  and  so  do  not  exert  a  pressure  that 
is  uniformly  distributed  circumferentially.  Any  concentrated  stress, 
such  as  that  in  the  concrete  supporting  a  rod,  diffuses  itself  in  the 
material  in  such  a  manner  that  its  intensity  rapidly  diminishes  with 
the  distance  from  the  surface  of  the  rod,  in  accordance  the  same  law 
as  exists  in  case  of  centers  of  attraction.  Since  the  supporting  com- 
pression under  the  rods  is  vertical,  we  can  imagine  the  lines  of  great- 
est compression  in  the  concrete,  when  this  component  is  combined 
with  those  already  mentioned,  to  lie  in  vertical  planes  on  a  bowl  or 
saucer-shaped  surface  which  has  as  many  indentations  or  scollops 
around  its  edge  as  there  are  radial  rods,  at  which  indentations  the 
slope  of  the  sides  is  such  more  nearly  vertical  than  a  slope  of  45°. 
At  such  parts  of  the  surface  the  intensity  is  also  more  severe,  and 
especially  is  this  the  case  if  the  slab  is  thin  so  that  the  concentrated 
pressure  has  small  opportunity  to  distribute  itself  by  radiating  into 
a  considerable  body  of  material  before  it  reaches  the  bottom  of  the 
slab.  It  thus  comes  about  that  thick  slabs  are  enabled  to  carry 
safely  larger  intensities  of  shearing  stress  around  the  cap  than  can 
thin  slabs,  which  is  in  accordance  with  and  in  justification  of  the 
statements  already  made  as  to  permissible  shears  around  the  cap. 

The  resulting  surface  of  fracture  due  to  shear  and  compression 
around  the  cap  would  be  of  irregular  conical  shape  starting  from 
the  edge  of  the  cap  and  extending  thru  the  entire  thickness  of  the 
slab,  were  this  not  interfered  with  in  the  upper  part  of  the  slab  by 
the  mat  of  reinforcing  rods,  which  are  so  tenacious  as  to  tear  to 
pieces  and  fracture  the  upper  surface  to  a  considerable  distance  in 
all  directions  whenever  any  such  fracture  occurs  around  the  column. 

Nevertheless  such  fracture  as  here  described  does  not  under 


RADIAL   AND    RING    RODS    PREVENT   FRACTURE    OF   HEAD  55 

any  ordinary  circumstances  result  in  a  dangerous  collapse  of  the 
slab,  or  one  that  cannot  be  repaired  without  much  difficulty,  for,  the 
radial  rods  and  the  reinforcing  rods  will  at  most  have  suffered  some 
individual  deformation  by  bending  and  are  still  far  from  being 
broken.  This  will  become  evident  later  where  an  experimental 
attempt  to  load  a  full-sized  slab  to  failure  is  described  in  detail,  and 
full  account  of  the  results  reached  is  explained  and  illustrated. 

It  is  stated  on  good  authority  that  in  experience  with  many 
hundreds  of  buildings  constructed  on  this  system,  no  case  of  shear 
failure  or  even  of  incipient  shear  failure  or  fracture  has  occurred  in  a 
well  cured  slab  near  the  column  and  while  a  few  cases  of  incipient 
failure  have  occurred  in  floors  where  forms  were  prematurely  re- 
moved, no  injury  or  fatality  has  resulted  therefrom  to  any  person. 

It  appears  that  the  line  of  weakest  section  in  the  cured  slab  of 
the  standard  mushroom  type  is  that  discussed  previously  in  obtain- 
ing (37)  and  shown  in  Fig.  3  page  7.  This  is  brought  out  later  by  a  test 
to  destruction  of  a  fairly  well  cured  slab.  The  line  of  weakest  sec- 
tion in  a  partly  cured  slab  is  on  the  other  hand  not  definitely  fixed, 
but  may  be  and  sometimes  is,  shearing  weakness  near  the  column  as 
has  been  discussed  and  pointed  out.  Provision  against  such  weak- 
ness or  carelessness  is  a  safeguard  which,  while  costing  a  small 
amount  in  the  matter  of  steel,  is  an  insurance,  against  serious  acci- 
dent well  worth  the  investment  involved.  It  is  secured  by  making 
the  radial  and  ring  rods  sufficiently  stiff  and  strong. 

15.  This  section  will  be  devoted  to  a  consideration  of  the 
mushroom  system,  and  to  several  more  or  less  similar  flat  slab 
systems,  in  order  to  comment  on  the  modifications  in  mechanical 
action  that  are  produced  by  the  particular  modifications  of  the 
arrangement  of  the  reinforcement  in  these  systems. 

Fig.  1,  page  2  represents  the  section  of  a  standard  mushroom 
head  by  a  vertical  plane  thru  the  axis  of  the  column.  In  this  the 
elbow  rods  are  shown,  the  vertical  portions  of  which  are  embedded  for 
such  distances  as  may  be  necessary  in  the  columns  or  are  them- 
selves column  rods.  One  of  these  is  represented  separately  at  the 
right  side  of  the  Fig.  They  are  confined  just  under  the  elbow  at 
the  top  of  the  column  by  a  steel  neck  band,  and  are  bent  over  at 
the  elbow  to  extend  radially  into  the  slab.  This  bent  over  portion 
is  formed  to  scale  as  to  length  and  slopes  in  accordance  with  the 
size  and  thickness  of  the  slab  in  which  it  is  to  be  used,  in  such  a 
way  that  when  the  ring  rods  and  four  layers  of  slab  rods  rest 
upon  it  and  are  tied  in  place,  the  top  of  the  upper  layer  will  be 


56  STANDARD   MUSHROOM    SYSTEM 

0.75  inch  below  the  top- of  the  slab  at  a  distance  of  the  thickness 
of  the  slab  outside  the  edge  of  the  cap,  and  at  the  same  time  the 
extremities  of  the  radial  rods  will  be  0.5  inch  above  the  bottom  of 
the  slab.  In  order  to  accomplish  this,  the  radial  portions  of  these 
rods  must  be  nearly  horizontal  over  the  cap,  and  have  a  suitable 
slope  outside  the  cap  as  shown  in  Fig.  1. 

Fig.  3,  page  7,  shows  the  ground  plan  of  the  reinforcement  of  the 
mushroom  slab  when  the  panel  is  square  so  that  L\  =  1/2  =  2a 
=  26.  In  this  Fig.  the  diameter  of  the  mushroom  head  is  assumed 
to  be  of  the  extreme  size  g  =  L/2,  a  size  which  would  increase  the 
cantilever  beyond  that  in  usual  practice  to  an  extent  not  adopted 
except  in  the  case  of  very  unusual  intensity  of  loading.  It  will 
be  observed  that  the  areas  where  the  reinforcement  consists  of  a 
single  belt  or  layer  are  thereby  rendered  small,  and  the  slab  action 
due  to  the  mutual  lateral  action  of  belts  which  cross  each  other 
exists  over  nearly  the  whole  slab. 

In  Fig.  2,  the  dimensions  of  the  rectangular  sides  are  so  taken 
that  Z/i/Z/2  =  0.75,  which  is  assumed  to  be  the  limiting  or  smallest 
value  of  that  ratio  for  constructional  purposes.  Further,  the 
diameter  of  the  mushroom  is  made  as  small  as  will  permit  the  rein- 
forcing belts  to  cover  the  entire  panel,  viz.  g  =  7  (a  +  6) /1 6.  For 
example  if  LI  =  20,  and  L2  =  15,  we  have  g  =  7.65+.  This  may 
be  considered  to  represent  standard  practice,  where  the  edges  of 
the  diagonal  belts  intersect  on  the  edges  of  the  side  belts.  This 
was  the  case  assumed  for  treatment  in  deriving  the  formulas  of  the 
preceeding  investigations.  Those  formulas  could  be  modified  to 
apply  to  larger  values  of  g,  by  taking  lines  of  contra-flexure  at  the 
edges  of  the  head  nearer  the  panel  center  than  given  by  (24),  and 
by  taking  larger  values  of  the  effective  cross  section  of  steel  than 
those  employed  in  (32),  (40)  and  (51). 

Now  it  is  evident  that  systems  similar  to  this  may  differ  from 
it  in  several  ways: — 

1st.  The  design  of  the  frame-work  at  the  top  of  the  column 
may  be  different  from  this  without  any  change  in  the  belts  of  re- 
inforcing rods.  It  is  hardly  possible  for  any  other  form  of  frame- 
work to  be  substituted  for  this  which  will  exhibit  the  same  rigidity 
of  connection  between  it  and  the  column  as  do  the  elbow  rods 
embedded  in  the  column  and  bent  over  radially  in  the  slab  so  as 
to  make  the  column  and  slab  integral  with  each  other  by  means 
of  this  common  reinforcement.  Any  reduction  of  the  stiffness  of 
connection  between  column  and  frame-work  of  head  results  in  in- 
creased tipping  of  the  head  under  eccentric  loading  of  the  slab. 


OTHER   SYSTEMS 


57 


Fig.  4. 

Eccentric  loading  is  any  loading  of  one  panel  differently  from 
another.  Tipping  of  the  head  increases  some  deflections  at  the 
expense  of  others,  and  increased  stresses  in  some  of  the  reinforcing 
rods  at  the  expense  of  others,  and  so  requires  some  additional 
reinforcement.  Such  a  frame-work  is  illustrated  in  Fig.  4,  which 
merely  rests  upon  the  top  of  the  column  without  the  support  of 
metallic  connection  with  the  vertical  column  rods.  It  consequently 
affords  less  resistance  to  tipping  under  eccentric  loads  than  when 
stiffened  by  such  metallic  connection. 

2nd.  The  ground  plan  of  the  reinforcing  belts  may  remain  un- 
changed but  part  only  of  the  belt  rods  may  be  carried  at  the  top 
of  the  slab  over  the  column  head,  while  the  rest  of  them  are  carried 
thru  under  the  head  at  the  bottom  of  the  slab.  This  modification 
of  design,  when  a  sufficient  number  of  rods  go  over  the  head  to 
resist  the  negative  bending  moments  there,  is  very  uneconomical 
of  steel,  because  in  the  case  where  they  all  go  over  the  head,  it  is 
the  fact  that  altho  the  mean  tension  of  the  steel  is  not  so  great  as 
at  mid  span,  nevertheless,  by  reason  of  the  overlapping  of  the  belts 
in  crossing,  the  stresses  in  the  rods  at  the  top  reach  a  value  not 
much  less  than  at  mid  span,  and  cannot  be  safely  diminished  in 
number.  It  thus  appears  that  the  rods  carried  thru  on  the  bottom 
are  largely  superfluous.  Of  these  two  mats  of  rods  at  top  and 
bottom,  one  of  them  is  necessarily  in  tension  and  the  other  in  com- 
pression. But  it  is  a  mistake  to  use  steel  to  resist  compression 
when  concrete  can  be  better  used  for  this  purpose.  The  lower  mat 
is  superfluous  for  this  reason. 


58 


SMALL    HEAD.       TOP   AND    BOTTOM    BELTS 


Y  x^c'-Y       <'  /r- 

rt—t-J'          \\    Assumed  line  of         I          If 
1 1    ^Tr^~lVl4^  nicCximum  bending    (  1  I 

X         Vd^/          it  Theoretical  line  \         ^~~— 

(\  co^n^^  yssr*.  \ 


^0 

ij 

1?° 

ii 

iW  in  pounds  per  square  foot 

\   I  i  1  i  1  1  I  I  i  !  i  !  i  1  i  I  11 

1  1  1  1  1 

)         ^^                                     Diagonal  steel  in  two  lauers     .    .  .   .                  ""^j^ 

~KU-^I      y^ 

a  layers 

q,n 


poundi  per  linear  foot 


Fig.  5 


There  is  still  another  and,  if  possible,  more  serious  objection 
to  this  arrangement  of  rods  to  form  a  mat  or  double  layer  of  rods 
at  the  top  and  at  the  bottom  of  the  slab  near  the  columns.  This 
is  because  they  are  too  far  removed  from  each  other  in  the  slab 
for  the  elongations  of  the  steel  in  one  mat  to  be  resisted  by  lateral 
contractions  in  the  other.  The  reinforcement  does  not  therefore 
conspire  to  produce  the  slab  action  expressed  by  Poisson's  ratio, 
which  requires  that  the  interacting  steel  concerned  should  lie  approxi- 
mately in  the  same  zone  or  level. 

This  arrangement  is  illustrated  in  Fig.  5,  copied  from  Taylor 
and  Thompson's  Concrete  Plain  and  Reinforced,  p.  484.  In  this 
design  the  size  of  the  head  is  small  enough  to  reduce  the  width  of 
the  belts  so  greatly  that  not  only  are  the  areas  where  we  have  a 
single  layer  of  rods  on  the  plan  much  enlarged,  but  we  find  that 
nowhere  do  more  than  two  layers  lie  in  metallic  contact  with  each 
other,  and  the  areas  where  even  this  occurs  are  limited  to  one 
relatively  small  square  over  each  column,  and  one  of  equal  size 
at  the  middle  of  each  panel.  The  remaining  areas  are  subject  to 
the  law  of  single  rod  reinforcement,  where  we  must  assume  lateral 
action  to  be  such  as  greatly  to  diminish  K  for  the  combination,  a 
fact  very  injurious  to  the  efficiency  of  the  reinforcement.  This 
as  has  been  said,  is  due  partly  to  the  smallness  of  the  head  and 
partly  to  the  separation  of  the  layers  between  the  top  and  the 
bottom  of  the  slab. 


BAD  EFFECT  OF  ANY  SHARP  BEND  OB  ELBOW  IN  A  ROD 


59 


3rd.  Another  modification  of  design  without  change  of  ground 
plan  is  that  where  the  rods  that  are  carried  over  the  head  at  the 
top  of  the  slab  are  given  a  sudden  steep  dip  at  the  line  of  contra- 
flexure  to  carry  them  to  the  bottom  of  the  slab  at  that  line.  This 
is  also  illustrated  in  Fig.  5.  Such  sudden  bends  or  kinks  any- 
where in  the  rods  may  give  rise  to  very  serious  fractures  because 
of  straightening  out  under  tension,  especially  when  the  forms  are 
removed.  Such  bends  give  rise  to  great  differences  of  stress  in  the 
extreme  fibers  of  the  rods,  thus  diminishing  their  resistance  also. 
All  sudden  bends  in  rods  embedded  in  concrete  should  be  sedulously 
avoided  as  tending  very  effectively  to  crack  the  concrete,  whether 
the  rods  are  part  of  the  belts  or  in  the  frame-work  of  the  head,  as 
shown  in  Fig.  3,  in  which  are  many  such  angles  and  elbows  unsup- 
ported except  by  concrete,  and  therefore  objectionable. 

It  seems  fair  to  conclude  that  the  cracks  shown  in  the  plan 
of  the  floor  of  the  Deere  &  Webber  Company  Building,  Minnea- 
polis, tested  by  Mr.  Arthur  R.  Lord,  and  occuring  along  the  edges 
of  some  of  the  loaded  panels  at  the  upper  surface,  where  none  usually 
appear,  were  due  to  the  elbows  in  the  frame  work  of  the  head,  like 
that  in  Fig.  4,  in  conjunction  with  the  comparatively  small  resis- 
tance to  bending  in  a  vertical  plane  offered  by  the  rods  forming  this 
projecting  elbow. 

In  the  mushroom  head  the  only  bend  permitted  is  that  at  the 
elbow  of  the  radial  rods  where  a  strong  steel  neck  band  prevents 
any  such  bad  effect  as  has  just  been  pointed  out. 


Fig.  6 


60  TWO  WAY  REINFORCEMENT 

4th.  We  may  notice  a  form  of  design  in  which  the  diagonal 
belts  are  omitted  and  the  entire  panel  is  covered  by  rods  parallel 
to  the  sides  of  the  panel.  This,  while  apparently  very  different  in 
ground  plan  from  those  just  considered  does  not  differ  from  it 
materially  in  principle.  It  is  clear  that  the  lattice  pattern  of  the 
web  in  this  case  is  in  many  parts  of  the  panel  not  woven  so  close 
as  where  diagonals  exist,  while  in  other  parts  of  the  mesh  the  num- 
ber of  layers  in  contact  with  each  other  has  been  decreased.  Experi- 
mental results  do  not  as  yet  enable  us  to  determine  with  certainty 
whether  Poisson's  ratio  for  this  combination  is  as  great  as  for  the 
mushroom.  Upon  that  depends  in  part  the  relative  efficiency 
of  the  two  arrangements.  A  form  of  this  design  is  seen  in  Fig.  6. 

The  maximum  deflections  at  the  center  of  a  loaded  panel  of 
the  system  of  Fig.  6,  would  occur  when  the  panels  touching  its 
four  sides  were  also  loaded.  In  this  particular  it  differs  from  a 
loaded  panel  in  a  mushroom  slab  which  would  theoretically  have 
its  deflection  slightly  decreased  by  loading  surrounding  panels, 
tho  this  is  too  insignificant  to  have  been  observed  as  yet. 

Deflections  shown  by  tests  of  this  system  of  two  way  reinforce- 
ment are  wholly  inconsistent  with  simple  beam  theory,  and  can 
only  be  explained  on  the  basis  of  slab  theory.  Nevertheless,  some 
of  its  advocates  attempt  to  design  its  reinforcement  and  com- 
pute its  strength  on  the  basis  of  beam  theory,  which  actual  de- 
flect ions  show  to  be  untenable.  Such  attempts  should  be  entirely 
abandoned  as  erroneous  and  misleading. 

All  considerations  which  have  been  discussed  under  the  three 
previous  counts  are  to  be  taken  as  applying  equally  to  this  plan 
of  arranging  the  reinforcing  rods,  especially  as  to  carrying  of 
part  of  the  belts  thru  on  the  bottom  surface  at  columns. 

5th.  Another  element  of  design  is  the  relative  number  of 
rods  in  the  side  and  diagonal  belts.  We  have  previously  adduced 
reasons  to  show  that  in  a  square  panel  the  same  number  of  rods  is 
required  ultimately  in  the  diagonal  belts  as  in  the  side  belts,  tho 
for  stresses  less  than  the  yield  point  of  the  steel,  it  would  be  pos- 
sible to  diminish  the  number  of  rods  in  the  diagonal  belts  some- 
what. Equation  (34)  shows  that  for  equal  stresses  in  the  steel 
of  the  side  belts  the  number  of  rods  should  have  the  same  ratio 
as  the  lengths  of  the  sides. 

A  different  rule  from  this  has  been  erroneously  proposed, 
viz.,  that  the  ratio  of  the  number  of  rods  in  the  side  belts  should 
be  equal  to  the  ratio  of  the  cubes  of  their  lengths.  The  only  foun- 
dation for  this  rule  is  that  according  to  the  beam  strip  theory  as 


RELATIVE   CROSS   SECTION   OF  BELTS  61 

developed  in  Marsh's  Reinforced  Concrete,  p.  283,  a  rectangular 
plate  carried  by  a  level  rigid  support  around  its  perimeter,  would 
divide  the  load  per  unit  of  area  which  is  carried  by  two  unit-wide 
rectangular  strips  that  cross  each  other,  as  the  fourth  power  of  their 
lengths,  and  hence  would  carry  to  the  edges  of  the  rectangle  loads 
proportional  to  the  cubes  of  the  lengths  of  those  edges.  Were  this 
so,  the  case  of  a  horizontal  rigid  support  around  the  entire  peri- 
meter of  the  panel  is  wholly  different  from  support  on  columns 
at  the  corners,  and  such  a  rule  would  be  wholly  inapplicable  there- 
fore to  a  floor  slab  so  supported.  This  rule  was,  however,  evidently 
adopted  in  the  design  of  the  Larkin  Building,  Chicago,  as  shown 
by  a  photograph  of  its  reinforcement  in  place  before  the  concrete 
was  poured,  to  which  the  writer  has  access  and  published  in  Cement 
Era  for  February,  1913.  The  very  exhaustive  tests  of  this  build- 
ing made  by  the  Concrete  Steel  Products  Company  of  Chicago, 
and  published  in  the  Cement  Era,  for  January  1913,  show  that  this 
ratio  of  rods  caused  the  stresses  for  the  larger  loads  to  be  more 
than  twice  as  great  at  the  middle  of  the  short  side  belts  as  at  the 
middle  of  the  long  side  belts.  This  was  assuredly  an  uneconomical 
distribution  of  steel,  since  correct  design  would  require  these  stresses 
to  be  equal,  when  in  fact  one  exceeded  the  other  by  120  to  140  per 
cent.  This  discrepancy  would  be  largely  rectified  by  making  the 
number  of  rods  directly  proportional  to  the  lengths  of  the  sides, 
as  required  by  (34). 

It  also  appears  that  the  diameter  of  the  mushroom  head  and 
the  width  of  belts  of  slab  rods  in  the  Larkin  Building  is  less  than  the 
limiting  size  in  the  standard  mushroom  system,  viz.  g  =  7(a+6)/16. 
This  makes  the  intersection  of  the  diagonal  belts  fall  nearer  the 
center  of  the  panel  than  the  edges  of  the  side  belts.  The  very 
considerable  effect  of  a  very  inconsiderable  change  of  this  width 
has  been  mentioned  on  p.  25.  The  result  would  be  that  the  steel 
would  for  this  reason  be  far  less  effective,  and  its  resistance 
would  be  more  nearly  in  accordance  with  (37)  than  with  (34) ,  a  loss 
of  perhaps  25  to  30%  in  its  effectiveness. 


62  SPECIMEN   COMPUTATION   OF   THIN   SLAB 

16.  This  section  will  be  devoted  to  a  specimen  computation 
applying  several  of  the  preceeding  formulas  to  a  floor  slab  of  practi- 
cally the  same  dimensions  and  reinforcement  as  one  or  two  recently 
designed  and  now  under  construction  (1913). 

Long  Side  LI  =  28'  X  12  =  336". 

Short  side  L2  =  25'  10"  =  310". 

Thickness  of  rough  slab,  h  =  10"  =  L/33.6. 

By  (56)  C2  =  IV/LI  =  0.9  nearly. 

Diameter  of  head  g  =  7  (Lx  +  L2)/32  =  141". 

Diameter  of  cap  LI— B  =  0.2^    =  67".     B  =  O.SLj  =  268.8". 

Each  belt  has  25  —  7/16"  round  rods. 

Cross  section  of  each  belt,  A  =  25  x  0.15+  =   3.76  sq.  inches. 

Depth  of  center  of  mid  side  belt  with  \  inch  concrete  cover- 
ing, di  =  10  —  0.5  —  0.2  =  9.3". 

Depth  of  center  of  second  layer  slab  rods  at  panel  center, 
dz  =  10  —  0.5  —  0.64  =  8.86" 

Depth  of  bottom  surface  below  third  layer  of  slab  rods  at  edge 
of  cap  with  %"  covering,  d3  =  10  —  0.75  —  1.1  =  8.15". 

Design  load  per  square  foot  =   150  Ibs. 

Dead  load  per  square  foot  =  130  Ibs. 

Panel  load,  W  =  280  x  28  x  25  5/6  =  202,550  Ibs. 

A  maximum  tension  is  found  in  the  slab  rods  at  the  middle  of 
the  long  side  belt,  and  is  to  be  computed  from  (34)  as  follows: 

202550  x  336 

/s=  -  -  =  11,120  Ibs.  per  sq.  inch (80) 

175  x  9.3  x  3.76 

Any  other  loading  within  elastic  limits  of  the  steel  would 
produce  proportionate  stresses. 

The  tension  in  the  steel  at  the  center  of  the  panel  is  com- 
puted by  (52),  as  follows: 

1.02  x  202550  x  336 

fs=  -  -    =  9,145  Ibs.  per  sq.  in..  (81) 

256  x  3.76  x  0.89  x  8.86 

The  radial  tension  at  the  edge  of  the  cap  is  by  (43), 

202550  x  336  x  646  (3  x  0.64  —  1) 

fs  =  -  -=  5320  Ibs.  per  sq.in .  (82) 

800  x  8.15  x  3.76  x  310 


COMPUTED    STRESSES    IN    THIN    SLAB  63 

The  circumferential  tension  at  the  vertical  section  thru  the 
center  of  the  column  at  the  end  of  the  long  side  may  be  computed 
by  placing  B  =  LI  in  (43),  and  we  obtain, 

202550  x  336  x  646  x  2 

/„=  -  -  =  11,570  Ibs.  per  sq.  in. .  (83) 

800  x  8.15  x  3.76  x  310 

as  the  mean  computed  intensity  of  stress  in  each  of  these  rods, 
regardless  of  its  distance  from  the  center  of  the  column.  This 
stress  may  be  reduced  by  increasing  the  number  of  laps  over  the 
head.  The  result  in  (83)  is,  however,  an  over-estimate  of  the 
tension  across  the  top  of  the  head  because  the  head  is  integral 
with  the  cap  of  the  column  where  compressions  in  the  concrete  are 
no  longer  confined  merely  to  the  thickness  of  the  slab  but  take 
in  a  much  greater  depth  of  concrete  in  the  cap.  This  in  effect 
puts  the  neutral  surface  at  a  lower  level  throughout  the  cap  and 
by  thus  increasing  the  lever  arm  of  the  reinforcement  reduces  its 
tension  and  deformation.  This  will  react  upon  the  rest  of  the 
reinforcement  in  such  a  manner  as  practically  to  make  the  stresses 
smaller  than  given  by  (83)  because  the  mean  lever  arm  will  have 
increased.  In  fact  the  greatest  stress  in  these  rods  will  be  that 
given  by  (80),  instead  of  (83). 

The  compression  in  the  concrete  lengthwise  of  the  longer 
side  belt  at  its  middle  is  to  be  computed  from  (38)  and  (80)  as 
follows:  By  taking  the  percentage  of  belt  reinforcement  at  0.3%, 
the  corresponding  value  of  i  =  0.72,  and  ES/EC  =  15: 

0.28x11120 

/c=  -  -   =  288  Ibs.  per  sq.  in (84) 

0.72  x  15 

The  compression  at  the  center  of  the  panel  where  the  per- 
centage of  slab  reinforcement  may  be  conservatively  assumed  at 
0.6%  and  i  =  0.66  may  be  computed  thus: 

9145 

/c=  -  -  =  305  Ibs.  per  sq.  in (85) 

2  x  75 

The  compression  at  the  edge  of  the  cap  lengthwise  of  the 
side  belt  is  uncertain  in  the  absence  of  exact  information  as  to  the 
laps  in  the  slab  rods  over  the  head.  Assume  that  one-half  the  rods 
are  lapped  over  each  head,  and  that  we  take  six  belts  as  the  reinforce- 
ment of  the  slab,  the  percentage  then  is  1.8%  and  i  =  J,  then, 

5,320 

/c=  -  =  355  Ibs.  per  sq.  in (86) 

15 


64  COMPUTED    DEFLECTION    OF    THIN    SLAB 

For  reasons  already  given  in  discussing  the  circumferential 
tensions  in  the  head,  it  appears  that  any  computation  of  the  cir- 
cumferential compressions  in  the  concrete  on  the  basis  of  (38)  would 
be  incorrect  and  subject  to  large  errors  of  as  much  possibly  as 
50%.  That  this  is  the  fact  appears  evident  when  we  consider  the 
large  mass  of  concrete  in  the  cap  which  must  be  actually  diminished 
in  lateral  dimensions  before  the  slab  which  is  integral  with  it  can 
be  subjected  to  true  stresses  of  equal  intensity,  and  consider  also 
that  near  the  edges  of  the  head  the  radial  rods  and  the  outer  ring 
rods  approach  the  lower  surface  sufficiently  to  afford  reinforcement 
to  resist  compression.  It  is  consequently  unnecessary  to  look 
further  than  (86)  in  •  computing  the  greatest  compression  in  the 
concrete. 

As  previously  stated,  computations  based  on  (38)  are  highly 
artificial  and  arbitrary  in  their  character,  since  they  assume  the 
straight  line  theory  as  well  as  an  arbitrary  value  of  the  ratio  of 
Young's  moduli  for  steel  and  concrete.  Furthermore,  concrete 
in  compression  in  both  circumferential  and  radial  directions  at  the 
same  time,  as  it  is  at  the  edge  of  the  cap,  is  known  to  resist  with 
safety  compressive  stresses  of  greater  intensity  than  when  in  simple 
compression  in  one  direction. 

If  a  test  load  of  twice  the  design  load,  viz.,  in  this  case  of  300 
Ibs.  per  square  foot,  be  placed  upon  the  slab,  the  deflections  which 
will  be  produced  by  the  addition  of  this  total  load  of  217,000  Ibs. 
may  be  computed  as  follows: 

217000  x  3363 

By  (54),      A  zi  =  -  -    -  0.237 (87) 

10.7  xl010x9.32x  3.76 

0.89  x  217000  x  3363 

By  (55),      Az2=-  =  0.378 ....  (88) 

6.56  xl010x8.862x  3.76 

217000  x  3363  x  2.084 

By  (58),      Az3=-  =  0.115.  ...  (89) 

60  x  1010  x  8.152  x  3.76 

0.89  x  217000  x  3363 

By  (60),      A  24  =  -  -7-  -  0.235 ....  (90) 

12.5  x!010x8.152x  3.76 


PROPORTIONATE  DEFLECTIONS  65 

By  (61),  Dl  =  0.352,  and  D2  =  0.613 (91) 

Dl  1  D2  1 

-,     and  - (92) 

LI          960  VLl2  +  L*         745 

Any  loading  differing  from  this  would  produce  deflections 
proportionate  to  its  intensity. 

In  this  specimen  floor  slab,  which  is  near  the  limit  of  least 
thickness  permissible  in  the  standard  mushroom  system,  viz., 
d  =  LI/ 35,  it  is  clear  that  the  design  load  brings  stresses  to  bear 
upon  its  reinforcement  which  are  very  moderate  in  their  intensity 
indeed.  It  is  also  evident  that  were  the  slab  to  be  loaded  with  a 
test  load  of  such  amount  that  the  total  load  sustained  would  be 
twice  the  dead  load  of  the  slab  itself  plus  twice  the  design  or  live 
load,  viz.  560  Ibs.  per  square  foot,  none  of  tjie  steel  would  be  stressed 
up  to  the  yield  point,  and  the  first  failure  would  take  place  by 
cracking  the  concrete,  tho  the  steel  would  still  prevent  sudden 
failure  and  collapse.  Alt  ho  the  slab  is  relatively  so  thin  the  de- 
flections are  also  very  small  for  so  large  a  span. 

It  has  not  yet  been  so  generally  recognized  as  it  should  be 
that  a  thin  construction,  such  as  a  flat  slab  is,  should  not  be  ex- 
pected to  show  so  small  proportionate  deflections  as  is  required 
in  girders. 

The  observed  results  of  quite  a  number  of  tests  of  mushroom 
slab  floors  are  to  be  found  on  pp.  32  and  44  of  Turner's  Concrete 
Steel  Construction.  These  are  there  compared  with  results  com- 
puted according  to  Turner's  empirical  formula,  which  translated 
into  our  present  notation  has  been  reproduced  in  equation  (72). 
The  observed  and  computed  results  show  a  very  close  agreement. 
The  results  given  by  (72)  are  in  close  agreement,  as  has  been  seen, 
with  those  derived  from  (61). 

Some  of  these  test  slabs  present  peculiarities  of  reinforce- 
ment such  as  need  to  be  individually  considered  in  order  to  make 
exact  computations  of  their  deflections.  It  is  thought  that  the 
specimen  computation  already  given  will  afford  sufficiently  guidance 
in  the  methods  to  be  employed. 

Having  considered  the  stresses  and  deflections  of  a  slab  which 
is  near  the  minimum  thickness  for  the  standard  mushroom  system, 
viz.  1/1/35,  it  will  be  instructive  to  consider  a  specimen  or  two 
near  the  maximum  thickness  Z^/20. 


66 


STIFFNESS  OF  MUSHROOM  SLAB  BRIDGES 


Tischers  Creek  Bridge,  Duluth 


Test  of  Tischers  Creek  Bridge  with  30  ton  construction  cars,  each  loaded  with  20  tons  of  rails 
Deflection  less  than  one  twenty  thousandth  part  of  the  span 


SPECIMEN  COMPUTATION  OF  THICK  SLAB  67 

Take  for  example  the  bridge  over  Tischer's  Creek,  Duluth, 
shown  in  the  cuts  on  page  vn  and  page  66.  It  is  supported  on  three 
rows  of  columns  crossing  the  gorge,  at  a  distance  apart  of  27  feet 
center  to  center  of  columns,  the  two  street  car  tracks  being  over 
the  side  belt  that  lies  along  the  center  line  of  the  bridge  lengthwise. 
Each  of  these  rows  consist  of  six  columns  lengthwise  of  the  bridge, 
at  a  distance  apart  of  26  feet  from  center  to  center,  so  that 

L!  =  27  x  12  =  324" 

L2  =  26  x  12  =  312" 

The  size  of  the  mushroom  heads  and  width  of  the  belts  is  12  feet, 
which  is  in  excess  of  7  (Li  +  L2)/32  =  139  1/8"  =  11.6',  thus  giv- 
ing great  stiffness.  The  object  to  be  obtained  by  maximum  thick- 
ness and  large  head  is  to  secure  great  stiffness  and  so  reduce  vib- 
rations as  well  as  decrease  deflections.  There  are  twenty  9/16 
inch  round  slab  rods  in  each  belt,  or  a  total  cross  section  in  each 
belt  of  A!  =  5  square  inches  of  metal.  The  slab  is  15"  deep  at 
its  thinnest  part  at  the  gutter  on  each  side  of  the  roadway,  and 
the  steel  is  kept  down  to  that  level  throughout  the  slab,  altho  at 
the  crown  of  the  roadway  under  the  tracks  and  over  the  center 
row  of  columns  the  slab  is  5"  thicker,  or  20",  with  the  same  thick- 
ness over  the  side  rows  of  columns  where  the  sidewalks  are.  The 
mean  thickness  is  somewhat  in  excess  of  L2/20.  This  makes 
di  =  19"  for  the  short  side  belts,  di  =  17"  for  the  long  side  belts 
and  d3  =  14"  approximately  for  the  heads.  The  design  load  per 
square  foot  =  150  pounds.  The  dead  load  of  the  slab  per  square 
foot  =  300  pounds.  Hence  W  =  450  x  26  x  27  =  315,900  pounds. 
The  effective  cross  section  of  slab  steel  is  so  great  by  reason  of  large 
heads  that  instead  of  (34)  we  may  take 

W  L 

(34) 


200  di  Ai 

For  the  long  side  belt  this  gives  /s  =  6,033  pounds  per  square  inch. 
The  total  load  imposed  on  the  slab  might  be  made  six  times  as  great 
without  causing  the  steel  to  reach  its  yield  point,  and  the  live 
load  might  become  900  pounds  per  square  foot  without  causing  /s 
to  exceed  16,000  pounds. 

This  slab  was  tested  as  shown  in  the  cut,  page  66,  by  running 
two  construction  cars  loaded  with  20  tons  of  rails  each  over  the 
bridge  at  the  same  time  along  one  track  of  the  short  side  belt  26 
feet  long.  Weight  of  each  car  =  60,000  pounds.  Weight  of  rails 
40,000  pounds.  Total  weight  of  train  =  200,000  pounds  extend- 
ing over  several  spans.  The  deflections  was  too  small  to  be  dis- 
covered by  observations  with  level  and  rod.  It  is  useless  to  attempt 


68 


COMPUTATION    OF   THICK    SLAB 


to  compute  the  deflection  of  this  slab  under  the  test  load  because 
the  four  steel  rails  of  the  railway  tracks  across  the  bridge  were  so 
fastened  to  the  steel  cross  ties  which  were  embedded  in  the  con- 
crete as  to  make  the  rails  a  part  of  the  reinforcement  of  the  slab. 
They  furnish  a  cross  section  of  reinforcement  equal  perhaps  to 
7  A  i,  which  would  effectually  bar  the  application  of  our  deflection 
formulas  and  reduce  deflections  to  very  small  quantities. 

In  so  thick  a  slab  as  this  the  action  of  any  contemplated  load 
is  widely  distributed  by  the  slab  itself,  and  such  loads,  as  well  as 
all  shocks  and  vibrations  are  largely  dissipated  or  absorbed  by  the 
body  of  slab  itself  without  causing  observable  local  stresses  as  they 
do  in  steel  structures. 


VIEW  OF  REINFORCING  STEEL 
Flat  Slab  Bridge,  Denver,  Colo.  Spans  43  ft.  6  in.  Carries  Heavy  Interurban  Cars 


COMPUTED    STRESSES    AND    DEFLECTIONS  69 

The  Curtis  Street  bridge,  Denver,  Colorado,  is  one  of  four 
bridges  across  Cherry  Creek,  shown  by  the  cut  on  page  68,  con- 
structed on  the  mushroom  system  It  has  three  rows  of  three 
columns  each  crossing  the  stream,  the  middle  column  of  each  row 
in  mid  stream  with  spans  of  42  feet  between  columns  centers  length- 
wise of  the  bridge,  thus  obstructing  the  waterway  as  little  as  pos- 
sible. It  has  a  width  of  28  feet  between  column  centers.  The 
slab  is  17  inches  thick  at  the  gutters,  26.5  inches  at  the  sidewalks 
outside  the  gutters,  and  21 "  over  the  center  row  of  columns.  The 
sidewalk  is  stiffened  with  fourteen  3/8"  round  rods  lengthwise 
just  below  its  top  surface  as  supplementary  reinforcement,  and 
there  is  an  outside  parapet  giving  added  stiffness.  There  are 
also  three  stiffening  rods  24"  apart  across  the  bridge  midway 
between  columns.  There  are  three  ring  rods,  and  the  width  of  the 
belts  is  16'.  This  is  in  excess  of  7  (Lx  +  L2)/32  =  183.75"  =  15  5/16'. 
The  heads  are  exceptionally  stiff  each  having  twelve  1  3-8"  round 
radial  rods.  Each  belt  has  twenty-six  5/8"  round  rods,  hence 
A  i  =  26x0.3  =  8  square  inches  nearly. 

L!  =  42  x  12  =  504"    ,     L2  =  28  x  12  =  336". 

The  dead  load  per  square  foot  =  300  pounds. 

The  design  load  per  square  foot  =  150  pounds. 

W  =  450  x  42  x  28  =  529,200  pounds. 

di  =  20"  for  long  side  belt. 

Compute  the  stress  in  the  steel  by  (34)  modified  to  (34) '  by 
reason  of  exceptional  stiffness,  and  we  obtain  /s  =  13,320  pounds. 

Compute  the  central  deflection  due  to  a  test  load  of  100  pounds 
per  square  foot.  Let  d3  =  16".  Then  in  (71)  L2/L1  =  2/3:  hence 
C2=  3/4,  and  we  have  D2  =  0.125".  This  is  probably  considerably 
in  excess  of  the  correct  deflection,  since  the  slab  is  stiffer  than  the 
one  considered  in  equation  (71),  which  was  derived  for  20  foot  spans. 
More  correct  values  are  to  be  computed  from  (54),  (58)  and  (61). 
Moreover  for  such  comparatively  light  stresses  in  the  concrete, 
the  deflections,  as  we  have  seen  previously  fall  short  of  those  com- 
puted by  the  formula,  which  agrees  with  experiment  for  stresses 
nearer  the  yield  point  of  the  steel.  D2  =  0.125"  is  less  than 
one  four-thousandth  of  the  span,  and  the  deflection  under  the 
working  load  would  undoubtedly  be  less  than  one  sixth-thousandth 
of  the  span. 


70  WORKING  STRESSES  AND  FACTOR  OF  SAFETY 

A  word  is  here  in*  place  respecting  working  stresses  and  the 
factor  of  safety  in  the  reinforcement  of  slabs,  to  the  effect  that 
the  same  values  of  these  quantities  in  slabs  affords  a  greater  degree 
of  security  than  in  ordinary  structural  steel  construction,  and  that 
occurs  for  several  reasons: 

1st.  Steel  rods  such  as  are  used  in  slabs  have  a  higher  yield  point 
by  perhaps  25%  than  the  steel  of  other  structural  members.  Fur- 
thermore, it  is  quite  possible  and  desirable  to  use  a  higher  carbon 
steel  for  these  rods  than  the  mild  steel  necessarily  used  in  structural 
work,  where  it  must  be  manipulated  in  such  ways  that  high  carbon 
steel  cannot  be  used.  But  in  these  rods  which  suffer  no  usage 
tending  to  impair  their  condition,  there  is  good  reason  to  use  a  steel 
of  higher  yield  point  and  greater  ultimate  strength.  This  yield 
point  may  readily  be  70%  greater  than  that  of  ordinary  mild  steel 
for  structural  purposes. 

2nd.  Rods  embedded  in  concrete  do  not  yield  as  do  bare 
single  rods  in  a  testing  machine  or  elsewhere  by  the  formation  of 
a  neck  and  drawing  out  at  that  point.  The  concrete  embedment 
prevents  that. 

3rd.  In  a  reinforcement  consisting  of  multiple  parallel  rods 
acting  together,  no  single  rod  can  become  overstrained  and  yield  to 
any  appreciable  extent  before  bringing  into  play  adjacent  rods. 
This  makes  the  construction  tough,  and  not  liable  to  sudden  col- 
lapse, as  well  as  obviates  concentration  of  stresses  thus  ensuring 
a  high  degree  of  security. 


COMPARATIVE   TEST   OF   TWO   SLABS  71 

17.  This  section  will  be  devoted  to  a  detailed  consideration 
of  a  test  to  destruction  of  two  slabs,  12'  x  12'  between  column 
centers,  constructed  for  experimental  purposes.  The  tests  were 
made  by  Professor  Wm.  H.  Kavanaugh,  in  November  and  December, 
1912,  and  the  results  he  obtained,  together  with  a  mathematical 
discussion  based  upon  them,  will  be  here  given.  One  slab  was 
constructed  in  accordance  with  the  plans  and  specifications  of  the 
U.  S.  Patent  No.  698,542  issued  to  O.  W.  Norcross  for  a  slab  for 
flooring  of  buildings,  and  the  other  was  a  Turner  Mushroom  slab 
under  U.  S.  Patent  No.  1,003,384.  The  test  serves  to  bring  out  in 
a  striking  manner  not  only  how  two  slabs,  which  present  a  super- 
ficial resemblance  in  the  plan  of  arrangement  of  reinforcement, 
differ  from  an  experimental  and  practical  standpoint,  but  it  also 
makes  evident  their  radical  divergence  of  action  mechanically  and 
mathematically. 

That  two  slabs  of  the  same  span,  thickness  and  amount  of 
reinforcement  should  on  test  show  that  one  of  them  was  more  than 
twenty  times  as  stiff,  and  more  than  five  times  as  strong  as  the 
other,  and  that  the  failure  of  the  weaker  one  was  a  sudden  and 
complete  collapse,  with  little  or  no  warning  to  the  inexperienced 
eye,  while  the  other  gave  way  by  slowly  pulling  apart  little  by 
little,  thus  gradually  getting  out  of  shape  without  any  final  break 
down,  are  phenomena  that  deserve  the  close  attention  of  the  de- 
signer, and  are  of  the  highest  interest  scientifically  as  well  as  practi- 
cally. The  enormous  differences  in  the  deflections  and  in  the 
stresses  in  the  reinforcement  as  shown  by  extensomoter  measure- 
ments, and  in  the  character  of  the  failure  in  respect  of  safety  and 
its  relation  to  the  line  or  zone  of  weakest  section,  as  well  as  in  the 
difference  of  design  loads  and  breaking  loads  amounting  to  500%, 
all  illustrate  what  scientific  design  will  accomplish  and  what  results 
are  possible  by  an  ingenious  arrangement  of  the  reinforcement. 

These  slabs  were  each  of  the  same  thickness,  viz  6",  and  were  sup- 
ported by  columns  placed  at  the  corners  of  a  square  12'  x  12'  from 
center  to  center  of  columns.  The  slabs  projected  2'  to  3'  beyond 
the  centers  of  the  columns  on  each  side,  and  had  precisely  the  same 
number  and  size  of  reinforcing  rods  in  each  belt,  viz  eleven  3/8 
inch  round  rods.  The  concrete  was  of  a  1  :  2  :  4  mix,  and  while 
only  about  four  weeks  old  at  the  time  of  the  test,  it  had  been  poured 
warm  and  kept  warm  by  steam  heat  under  such  unusually  favorable 
conditions  as  to  have  become  well  cured  at  the  time  of  the  test. 
The  steel  used  showed  by  test  a  stress  at  yield  point  of  51,000  to 
55,000  pounds  per  square  inch,  and  an  ultimate  strength  of  76,000 


72  BEAM    THEORY,    VERSUS    SLAB    THEORY 

to  80,000  pounds,  with -an  elongation  of  twenty  to  twenty-five  per 
cent. 

The  first  slab  was  made  in  accordance  with  the  specifications 
of  the  Norcross  patent  already  referred  to  except  that  belts  of  rods 
were  substituted  for  the  netting  mentioned  by  the  patentee.  This 
design  was  selected  as  one  of  the  two  for  this  comparative  test, 
not  because  it  is  a  good  design,  or  one  that  any  engineer  would 
to-day  care  to  employ,  but  because  it  exhibits,  according  to  the 
express  intention  of  the  patentee,  simple  tension  on  its  lower  surface, 
everywhere  between  columns,  and  simple  compression  everywhere 
on  its  upper  surface  between  columns;  this  being  in  direct  contrast 
to  the  other  design,  which  is  arranged  not  only  to  resist  direct  ten- 
sions over  the  supports,  which  the  first  does  not,  but  also  to  resist 
circumferential  stresses  both  around  the  supports  and  around  the 
panel  centers,  as  any  truly  continuous  flat  slab  must. 

This  test  may  then  be  viewed  in  the  light  of  an  experimental 
demonstration  of  the  difference  between  a  reinforced  flat  slab  con- 
structed in  accordance  with  the  beam  theory  and  one  constructed 
in  accordance  with  correct  slab  theory,  where  true  and  apparent 
moments  differ  radically  as  shown  at  the  beginning  of  this  investi- 
gation, but  are  wholly  contradictory  to  any  form  of  simple  or  con- 
tinous  beam  theory.  This  test  may  be  regarded  as  settling  once  for 
all  the  question  of  applying  simple  beam  theory  to  a  cantilever  flat 
slab,  reinforced  throughout  practically  its  entire  area  with  a  lattice  of 
rods  crossing  each  other  and  in  contact.  It  shows  that  it  is  impos- 
sible to  compute  the  deflections  of  such  a  slab  by  beam  theory. 
Furthermore  this  impossibility  makes  it  certain  that  the  stresses 
in  such  a  slab  cannot  be  computed  by  beam  theory,  for  to  do  this  is 
to  commit  an  inconsistency  such  as  has  heretofore  too  often  been 
committed,  but  one  which  should  hereafter  be  carefully  avoided. 


THE    NORCROSS   TEST    SLAB 


73 


Norcross  in  his  patent  already  referred  to  describes  his  con- 
struction as  consisting  "essentially,  of  a  panel  of  concrete  having 
metallic  network  encased  therein,  so  as  to  radiate  from  the  posts 

on  which  the  floor  rests The  posts  are  first  erected,  and  a 

temporary  staging  built  up  level  with  the  tops  of  posts.  Strips  of 
wire  netting  are  then  laid  loosely  in  place  on  top  of  the  staging .... 
The  concrete  is  then  spread  upon  or  moulded  in  place  on  the  staging 
to  enclose  the  metallic  network.  In  practice  I  have  sometimes 
laid  the  concrete  in  layers  of  different  quality,  the  lower  layer  of 
the  floor  which  encloses  the  wire  being  laid  with  the  best  concrete 

available If  the  forces  acting  upon  a  section  of  flooring 

supported  between  two  posts  be  analyzed  it  will  be  found  that  the 
tendency  of  the  floor  section  to  sag  between  its  supports  will  cause 
the  lower  layers  of  the  flooring  to  be  under  tension  while  the  upper 
layers  of  the  flooring  will  be  under  compression,  these  stresses  being, 
of  course,  the  greatest  at  the  top  and  bottom  layers,  respectively." 


Fig.  7.     Reinforcement  of  Norcross  Slab 


74 


THE    NORCROSS   TEST 


Fig.  8.     Norcross  Slab  Carrying  Load  3 
'-  o 


Col.  Cop  Phts  20*20*£ 

Be/fo  //~i  ^eoch 

Fig.  9.     Norcross  Slab 


LOADS    ON    MUSHROOM    SLAB 


75 


3 

"to 

CO 


r_C> 
3 


13 

1 

CO    CO    ^^    ^O    CO    t^    t^*    t^*    1^ 

rH 
00 

$ 

£ 

l^  ir^  oc  l^  t^ 

cr 

CO 

GOCXDCOOOOOOOOO 
O^l    O^-l    *-O    C^l    C^l    *-O    ^-O    ^-O    ^-O 

CO 
<M 

J^ 

(MCMTtiC^C^C^C^C^C^ 

1 

IS 

GO    GO    O    GO    GO 
GO    GO    (N    GO    GO 

S 

^ 

c^ 

H 

CO     CO     rH     CO     CO 
rH 

CO 

CO 

d 

-d 

l>-    !>•    GO    !>•    !>• 

cr 

CO 

s_ 

00    GO    CO    GO    GO             03 

CO 

a 

^ 

O 

•4-9 

GO    GO    C^    GO    GO             QJ 

3 

H 

CO    CO    CO    CO    CO 

CO 

02 

d 

$ 

F 

GO    00    O    00    GO 

c^ 

IH 

(N    (N    C^    C<1    C^ 

rH 

•a 

GO    GO    (M    GO    00 

j 

"Q 

CO    CO    O    CO    CO 

•^ 

H 

CO    CO    <M    CO    CO 

rH 

d 

t 

rH     rH     rH     rH     rH 

O 

co 

S, 

CO 

o 

o 

2 

^pqoQH^OWhH 

1 

76 


NORCROSS    SLAB 


The  number  and  arrangement  of  the  reinforcing  rods  in  the 
Nor  cross  experimental  slab,  (eleven  3/8"  round  rods  in  each  side 
and  diagonal  belt)  is  clearly  shown  in  the  view  of  Oct.  3 1st,  Fig.  7,  which 
shows  the  forms  ready  for  pouring  the  concrete.  Steel  plates 
20"  x  20"  x  0.5"  carry  the  rods  and  rest  on  the  tops  of  the  columns, 
which  last  in  this  case  consisted  of  steel  pipes  about  5J"  in  dia- 
meter filled  with  concrete  and  embedded  at  their  lower  ends  in  large 
concrete  blocks.  A  vertical  central  bolt  in  the  concrete  at  the 
upper  end  of  each  pipe  permitted  the  plates  to  be  firmly  secured  to 
the  tops  of  the  columns.  The  view  of  Nov.  30th,  Fig.  8,  clearly 
shows  the  manner  of  placing  the  pig  iron  on  the  slab  for  load  3. 
This  slab  is  16'  x  16'.  The  loading  at  first  covered  an  area  having 
the  form  of  a  Greek  cross  whose  central  square  was  five  feet  on  a 
side  with  arms  5'  6"  long,  as  represented  in  accompanying  diagram 
of  loaded  areas  A,  B,  C,  D,  E,  Fig.  9,  and  of  amounts  shown  in 
Table  1. 


Fig.  10.     Collapse  of  Norcross  Slab 

When  10,000  pounds  had  been  piled  on  the  central  part  of  the 
slab  in  addition  to  load  No.  4,  of  66,812  pounds,  the  slab  suddenly 
failed.  In  anticipation  of  such  failure  timber  blocking  had  been 
placed  under  the  slab  to  prevent  its  falling  more  than  possibly  ten 
or  twelve  inches. 


NORCROSS    TEST 


77 


Fig.  11.     Collapse  of  Norcross  Slab 

The  two  views  of  Dec.  2d,  Fig.  10  and  Fig.  11,  show  the  con- 
dition of  the  slab  after  removing  part  of  the  final  loading  in  order 
to  render  the  nature  of  the  failure  visible.  Careful  extensometer 
measurements  of  the  elongations  of  the  steel  rods  at  the  middle 
of  the  side  and  diagonal  belts  were  made  under  the  action  of  loads 
1,  2,  3  and  4,  and  also  similar  extensometer  measurements  in  the 
concrete  both  on  the  top  and  the  bottom  of  the  slab  along  the  center 
line  of  the  side  and  diagonal  belts  near  those  edges  of  two  of  the 
steel  plates  which  were  nearest  the  center  of  the  belts.  Besides 
these,  certain  other  measurements  of  the  concrete  were  made  at 
right  angles  to  the  diagonals.  Deflections  were  also  measured 
under  these  loads  at  the  middle  of  the  diagonal  belt  and  of  two  of 
the  side  belts  at  V,  W,  X,  Y,  Z. 

These  measurements  all  show  beyond  question  that  the  side 
and  diagonal  belts  act  like  simple  beams  in  this  form  of  construction, 
since  the  stresses  in  the  steel  and  concrete  on  the  under  side  of 
the  slab  in  the  direction  of  the  rods  is  invariably  tensile,  while  the 
stresses  in  the  same  directions  on  top  of  the  slab  are  always  com- 
pressive.  It  was  the  avowed  intention  of  Norcross  to  reinforce 
the  slab  in  this  manner  since  he  regarded  the  upper  part  of  the  slab 
as  being  subjected  everywhere  to  compression  and  the  lower  part 
to  tension  only,  as  stated  in  his  specifications  as  already  quoted. 


78  COMPUTATION    OF   THE    TEST 

The  following  computation,  Table  2,  shows  a  good  approxi- 
mate agreement  of  the  results  of  this  test  with  the  beam  theory  of 
flexure,  assuming  for  simplicity  that  the  stiff  steel  supporting  plate 
and  interlacing  of  the  ends  of  the  belts  diminishes  the  effective 
span  of  the  side  belts  by  12",  and  the  diagonals  in  the  same  pro- 
portion, and  further  assuming  that  the  loading  was  all  applied  at 
the  middle  of  the  side  and  diagonal  belts. 

The  extensometer  measurements  made  were  for  a  length  of 
8",  consequently  the  stress  in  the  steel  per  square  inch  would  be 
computed  thus: 

/s  =  l/8  (elongation  in  8")  x  30,000,000; (l)j 

and,  this  being  known  from  observation,  it  will  be  possible  to  com- 
pute the  load  W  carried  by  the  beam  in  which  the  given  elongation 
occurs,  as  follows: 

The  bending  moment  due  to  a  concentrated  load  W  at  the  mid- 
dle of  a  beam  of  length  L  is  M  =  J  W  L, (2)x 

and  the  equal  moment  of  resistance  of  the  reinforcement  by  which 

it  is  held  in  equilibrium  is    M=A  j  d  fs (3)i 

in  which  A  is  the  total  cross  section  of  the  steel  in  the  belt  = 
11  x  0.11  =  1.215  sq.  in.,  and  the  distance  from  the  center  of 
the  steel  to  the  center  of  compressive  resistance  of  the  concrete 
is  assumed  to  be,  j  d  =  0.9  x  5.75 

when  d  =  5.75  is  taken  as  the  distance  from  the  center  of  action 
of  the  steel  to  the  top  of  the  slab, 

Hence  W   =  4  A  j  d  fs/L .  .(4)i 

is  the  load  required  to  cause  the  stress  fs  in  the  steel.  In  the  side 
belts  we  assume  the  span  L  to  be  132",  and  in  the  diagonals  132  V2. 

In  Table  2,  which  follows,  it  will  be  noticed  that  loading  No.  1 
is  too  small  to  develop  sufficient  elongations  or  deflections  to 
overcome  the  initial  compressions  in  the  concrete  in  which  the 
reinforcement  is  embedded,  so  that  the  load  carried  by  the  steel  is 
only  about  one  half  of  the  actual  load,  the  other  half  being  evidently 
carried  by  the  concrete  in  which  it  is  embedded.  This  is  in  com- 
plete accord  with  other  similar  experiments.  But  in  case  of  loads 
No.  2  and  No.  3,  where  the  steel  is  stressed  close  to  the  yield 
point,  the  sum  of  the  loads  as  shown  by  the  stresses  in  the  steel 
is  very  close  to  the  total  actual  load.  It  is  assumed  that  these 
total  actual  loads  are  carried  by  the  various  belts  in  the  same  pro- 
portion as  the  computed  loads,  since  there  is  no  other  way  of 
dividing  the  total  load  between  the  belts.  This  may  be  stated 
mathematically,  as  follows: 


LOADS   AND    DEFLECTIONS    OF   NORCROSS    SLAB 


79 


")-H        J3 

rf)    tt 

It 

•^  o 

!tf- 
III 

IP 

S    Q^ 

(" 1  tS     fl3 


QQ'® 


^.2^ 

•s-s  ° 

CO     ^ 

§^ 


CO 


?? 


O    Th 


O5  O 

<N  Oi 

i— I  LO 

O  TjH 


to 

CO 


Tf    00    CO 

S£8 


8§ 

CD    "^ 
!>.    i— i 


00 


•§  -S 

1  a 

^        -H 

o  a 

Pn     O 


j£ 
« 


(M 


o  o 

!>•    to 

O    CD 


tO    CO 
OS    1-1 

i-H     CO 


(S 

&b 
s 


g§ 


(M 


!! 

11 


(N 


to  i^ 

oo  to 

10  CD 

CO  CO 


S3 

<M    CO 


00 


O 

0 


73     ^ 

11 


r°    r° 
H    H 


rh    to 
O    O 


O    !>•    1-1 

O    CO    TtH 


O    O 


00    00    (M 

g-- 

O    O 


^    0 


' 


73      g 


§1§ 

00        •        • 
TT    O    i— i 


O    <M    O 

CO    O    O 


O  to  <N 

rin  CO  <N 

O5 

rH  O  O 


"S 


PQ 


S 


80  NORCROSS  SLAB  ON  THE  BEAM  THEORY 

Let  Wi  =  the  computed  load  on  a  side  belt, 
and  W2  =  the  computed  load  on  a  diagonal  belt. 
Let  Wi  =  the  actual  load  on  a  side  belt, 
and  W2  =  the  actual  load  on  a  diagonal  belt. 
Then  4TFi  +  2TF2  =  total  computed  load  on  slab, 
and  4TFi  +  2TF2  =  total  actual  load  on  slab. 

4  Wi  +  2  W2        Wi        W2 

Then  -  -  =  -    -=--, (5)2 

4  Wl  +  2  W2        Wl        W2 


from  which  TFi  and  T72  can  be  computed,  W\  ,  W2  and  4W[  +  4TF2 
being  already  known. 

The  stresses  in  the  steel  under  load  No.  4,  are  so  far  beyond 
the  yield  point  as  to  make  computation  useless.  Having  found 
the  actual  distribution  of  loading  W[  and  W2  the  center  deflections 
of  the  belts  have  been  computed  by  simple  beam  theory  from  the 
formula. 

W'  L3 

D2=  -  — (6)! 

48  E  A  ijd2 

in  which  i  d  =  the  distance  from  the  steel  to  the  neutral  axis  and 
the  value  of  j  has  been  assumed  to  be  0.69;  W'  is  the  actual  load  on 
the  belt  and  L  is  its  span  as  previously  stated. 

It  appears  from  Table  2,  that  the  effect  of  the  reinforcement 
is  accounted  for  to  a  reasonably  close  approximation  by  consider- 
ing the  belts  to  act  as  a  combination  of  simple  beams,  at  least  with- 
in the  range  of  loading  near  the  yield  point  of  the  steel. 

It  appears  that  the  steel  reached  its  yield  point  under  a  total 
load  on  the  slab  of  from  15  to  18  tons  and  final  collapse  occured  under 
a  total  load  of  a  little  over  twice  the  latter  amount  not  distributed 
uniformly  but  piled  more  in  the  general  form  of  a  pyramid. 

It  was  observed  that  the  application  of  the  relatively  small 
loading  on  the  corner  areas  F,  G,  H,  I,  had  a  very  injurious  effect 
upon  the  slab,  tending  to  break  it  across  the  tops  of  the  columns. 

The  results  of  the  test  may  be  summarized  in  the  Norcross 
system  as  follows: 

1st.  This  slab  is  of  the  simple  beam  type,  and  the  test  shows 
no  cantilever  action  and  no  circumferential  slab  action. 

2nd.  The  narrow  belts  running  diagonally  leave  large  areas 
without  reinforcement,  and  there  is  consequently  no  provision  for 
resisting  circumferential  tensions  as  required  in  slab  action. 

3rd.  The  concrete  showed  compressive  stresses  on  the  upper 
surface  of  the  slab  in  the  direction  of  all  the  reinforcing  rods. 


SUMMARY    OF    NORCROSS    TEST  81 

4th.  The  concrete  showed  tension  at  the  bottom  surface  in 
the  direction  of  all  the  reinforcing  rods,  in  agreement  with  Norcross' 
own  analysis. 

5th.  This  slab  deflected  1.6"  under  33  tons  and  then  broke 
down  completely  under  38  tons. 

6th.  The  first  crack  appeared  under  a  load  of  15  tons  and 
deflection  of  0.7". 

7th.  The  slab,  not  being  reinforced  on  the  top  surface  over 
the  columns,  inevitably  cracks  at  a  column  when  the  slab  is  loaded 
around  the  column. 

8th.  At  failure  the  steel  had  passed  its  yield  point.  The 
percentage  of  reinforcement  in  the  diagonal  belt  if  we  regard  the 
belt  as  about  18"  wide  is  very  nearly  1%,  but  since  a  width  of 
concrete  somewhat  greater  than  that  may  be  assumed  to  act  with 
this  steel,  the  percentage  of  reinforcement  is  somewhat  less  than 
1%.  Similarity,  the  side  belts  of  width  36"  have  a  reinforcement 
less  than  0.5%.  The  full  strength  of  the  steel  in  both  belts  was 
developed  by  the  concrete,  which  fact  demonstrates  that  the  con- 
crete was  of  high  grade  and  well  cured.  The  steel  was  also  of 
good  standard  quality,  and  the  test  was  therefore  in  every  way 
fair  to  the  Norcross  slab,  since  it  was  so  loaded  as  to  cause  the 
stresses  in  the  side  and  diagonal  belts  to  be  practically  equal,  thus 
using  the  steel  most  economically.  The  slab  failed  because  the 
steel  yielded  near  the  middle  of  the  spans,  thus  causing  the  concrete 
above  the  steel  to  crack  and  break. 

The  second  slab  was  made  according  to  the  Turner  Mush- 
room System,  under  the  patent  already  referred  to. 

Since  all  forces  in  a  plane  may  be  resolved  into  components 
along  any  pair  of  axes  at  right  angles  to  each  other  it  is  possible 
to  provide  reinforcement  to  resist  any  horizontal  tensile  stresses 
in  the  slab  by  various  arrangements  of  intersecting  belts  of  rods  at 
zones  where  these  stresses  occur.  The  combination  of  such  belts  with 
radial  and  ring  rods  to  constitute  a  large  and  substantial  canti- 
lever mushroom  head  at  the  top  of  each  column  affords  a  very 
effective  and  economical  arrangement  for  controlling  the  distribution 
of  the  stresses  in  the  slab,  and  it  places  the  reinforcement  where 
it  is  most  needed.  It  not  only  has  the  same  kind  of  advantage 
that  the  continuous  cantilever  beam  has  over  the  simple  girder 
for  long  spans,  but  combines  with  it  the  kind  of  superiority  that  the 
dome  has  over  the  simple  arch  by  reason  of  circumferential  stresses 
called  into  play,  which  greatly  adds  to  the  carrying  capacity  of  the 
slab. 


82 


REINFORCEMENT   OF   MUSHROOM   TEST   SLAB 


Fig.  12.     Reinforcement  of  Mushroom  Slab 


Column   Rods  8~/8*  Diom. 

Fig.  13.     Mushroom  Slab 


THE    MUSHROOM    SLAB    TEST 


83 


The  mushroom  test  slab  was  six  inches  thick,  and  was  sup- 
ported on  four  18"  by  18"  square  reinforced  concrete  columns 
distance  12'  from  center  to  center.  These  had  square  capitals, 
42"  x  42".  The  slab  was  appromimately  18'  x  18',  and  the  dia- 
meter of  the  outer  ring  rod  of  the  Mushroom  was  66",  while  the 
inner  ring  was  42".  These  were  supported  on  eight  1-1/8"  round 
radial  column  rods. 


Fig.  14.     Mushroom  Slab,  Load  4. 

This  will  be  clearly  understood  from  the  view  dated  October 
31st,  Fig.  12,  which  shows  the  reinforcement  and  forms  ready  for 
pouring  the  concrete.  The  remaining  views  are  explained  by  their 
accompanying  legends. 

The  diagram  of  loaded  areas  for  the  mushroom  slab  Fig.  13,  is 
like  that  already  given  for  the  Norcross  slab  in  every  particular 
except  that  the  size  of  the  mushroom  slab  being  18'  x  18',  while  the 
Norcross  slab  was  16'  x  16',  the  arms  of  the  Greek  cross  in  the 
mushroom  slab  are  each  5'  6"  long  and  5'  wide. 


84 


LOADS    ON    NORCROSS    SLAB 


g 

CO 

•3 


CO 


1 

C<1    (N    CO   C3    CQ 
<M    (M    TJH    (M    <N 

CO    CO    »O    CO    CO 
O    O    T-H    O    O 
(M    (M    CO    C^    <M 

CO 

T-H 

tO 

u 

CO    CO    00    CO    CO 

(M    (M    CO    <N    <N 

CO    CO    <N    CO    CO 

00 

3 

18 

(M    <>>    CO    (M    (M 
(M    (M    TJH    (M    (M 

00    00    to    00    00 
CO    CO    T-H    CO    CO 
T-l     T-H     (M     rH     T-H 

to 

Tfl 

^j 

CO    CO    00    CO    CO 

CO 

<M    (M    CO    <M    (M 

CO 
(M 

1 

(M    (M    00    <M    (M            ^ 

to    tO    to    to    to            i-j-t 

to  to  -*  to  to         a 

CO    CO    O    CO    CO            § 

CO 

CO 

CD    **^ 

CO    CO    CO    CO   CO            § 

T-H        T-H        00       T-l       TH                          CO 

S  S  5  §  §          £ 

CO 

T-H 

r—  1 

13 

<N    (N    O    (N    (M            2 

to  to  -^  to  to 

tO    to    O    to    to             ^ 
CO    CO   to    CO    CO           -^ 

00 

cS 

T-H 

CO 

K  £ 

CO    CO    cO    CO    CO 

« 

a£ 

o  o  o  o  o 

(M    (M    (N    C^    (M 

% 

1 

cO    cO   C^    CO    CO 

<N    (M    to    <N    (M 

CO    CO    (M    CO    CO 

i 

T-H 

CO 

00    00    00    00    00 

SO    O    O    O 
O    O    O    O 

T—  I       T-H       T-H       T—  1        T-H 

00 

1 

1 

<5  pQ  O  P  W 

i 

LOADS    ON    MUSHROOM    SLAB 


85 


& 


O5COO2O5OOOO 


COO2O5OO 
OO(N<Mi-ii-H 


CO 

OS  OS  <M  OS  iO 
CO 


OOO 


8 


00 


| 

3 
O 
PH 


02 

i 


O 

h^ 


OS 


CO 


O    O 


CO 


s 

CO 

s 

(M 


00 


O    O    O    O 
^O    *-O    ^O    ^O 


CO 
CO 


(M(MCO(M<MCOCOCOCO 
T—  II-H<NT—  ii—  iiOiO^OiO 
lOiOOSiO^OOOOO 


OS  OS  T^  os  OS 


QO 


S 


JS 

2 


^ 


<M  (M  CO 


CO  CO  l>-  CO  CO 
<M  (N  CO  <N  (M 


CO  CO  O  CO  CO 
CO  CO  OO  CO  CO 


§ 

CO 
<M 

CO 


02 


86 


LOADS   ON   MUSHROOM    SLAB 


Fig.  15.     Mushroom  Slab,  Load  7 . 


Fig.  16.     Mushroom  Slab,  Load  9. 


COMPUTATION   OF    MUSHROOM   TEST  87 

The  accompanying  Table  3,  exhibits  the  loads  per  square  foot 
of  each  of  the  subsidiary  areas  shown  in  the  diagram  as  also  the 
total  loads  on  each  of  those  areas.  The  view  of  Dec.  3,  Fig.  14, 
shows  load  4,  and  that  of  Dec.  13,  Fig.  15,  load  7,  while  that  of 
Dec.  16,  Fig.  16,  shows  load  9. 

Elongations  of  steel  were  measured  by  Berry  extensometers 
in  two  of  the  side  belts  and  in  one  of  the  diagonal  belts  until  the 
yield  point  of  the  steel  was  reached  at  load  No.  8.  Deflections 
were  also  measured.  In  Table  4,  these  will  be  considered  so  far  as 
they  relate  to  the  middle  points  of  the  belts.  Loads  8,  9,  10,  are 
of  great  interest  as  exhibiting  the  behavior  of  the  slab  under  ex- 
cessive loads,  showing,  as  they  do,  yielding  and  large  permanent 
deformation  without  dangerous  collapse. 

By  (52)  the  uniformly  distributed  load  per  square  foot  of 
panel  area  when  the  stress  in  the  diagonal  belt  is  /„  is  found  for  a 
square  panel  from  the  expression 

256  j  d2  A 

-  — fs (52a) 

144  L 

which  applied  to  this  slab  gives  us 

256  X  0.89  X  5.125  X  1.215 

w   =  -  -/.  =/s/14.6 (52b) 

144  X  144 

The  values  of  this  uniformily  distributed  load  w  is  tabulated 
in  table  4,  for  each  of  the  observed  values  of  the  /„  in  the  diagonal 
belts.  The  values  of  w  so  computed  tend  to  become  identical, 
in  case  of  the  heavier  loads,  with  the  loads  per  square  foot  on  the 
central  area  C,  as  might  reasonably  be  expected,  w  being  the  uniformly 
distributed  load  which  is  equivalent  so  far  as  the  stress  on  the  dia- 
gonal belt  is  concerned  to  the  action  of  the  actual  loads  which  are 
not  uniformly  distributed. 

How  compute  by  (54),  (55),  (58),  (60)  and  (61),  the  deflections 
at  the  mid  side  belt  and  at  center  of  the  panel,  due  to  a  uniform  load. 
These  results  are  given  in  Table  4,  and  accord  closely  with  those 
actually  observed,  as  they  should,  because  the  irregularity  of  dis- 
tribution does  not  produce  deflections  that  differ  much  from  the 
equivalent  uniform  load  as  computed  above. 

In  these  computations  it  is  assumed  that  di  =  5.5",  d2  = 
5.125",  d3  =  4" 


88  DEFLECTIONS    OF    MUSHROOM    SLAB 

The  double  set  of  values  under  loads  4  and  5  is  due  to  the 
fact  that  readings  were  had  under  load  4,  immediately  after  the 
load  was  applied,  and  again  7  days  later  before  applying  load  5. 
The  second  set  of  readings  were  the  larger  as  shown.  The  second 
set  of  readings  under  load  5,  were  taken  four  days  subsequently 
to  the  first  set. 

It  appears  from  Table  4,  that  the  observed  results  are  account- 
ed for  by  the  slab  theory  to  a  good  degree  of  approximation 
up  to  the  yield  point  of  the  steel. 


Fig.  17.     Comparative  Deflections  of  Norcross  and  Mushroom  Slabs. 


A  graphical  representation  of  the  experimental  observations 
in  the  deflections  at  the  points  V,  W,  X,  Y,  Z,  of  the  two  slabs  is 
found  in  Fig.  17,  which  shows  in  a  striking  manner  how  small  the 
loads  and  how  great  the  deflections  were  in  the  Norcross  slab  on  the 
one  hand,  and  how  large  the  loads  and  how  small  the  deflections 
were  in  the  mushroom  slab  on  the  other  hand. 


LOADS    AND    DEFLECTIONS    OF    MUSHROOM    SLAB 


89 


II 
§| 

ll. 


§11 


pi 

•111 

1 

*S  «  S 

g  20 
o*-§ 

•  i— i     03 

•+3   hfi 


as 


I 


CO 


iO 


CO 


'S 


00 


CO    O 

l>-     rH 

CO    CO 


O    O 


00 
10 


fab  ^ 

^        Q) 

S  m 


S 


CO 

^H 

00 


O 

iO 

iO 

CO 


O  O 

rH  1O 

CO  • 


05 

CO 


O    O    00    O    O    OO    *O 

<N  rH 


OOCO(MrHlOrHrH 
»0     O  ' 


C9  '   C4  t» 

CO  OO  iO 

CO  rH  O 

(M  • 


LO 


Oi 


CO 

Tt< 

O 


CO 


CO 

O 


10        10  co 

|>.               •  O. 

O2                >O  (M 

CO  O 


OO  O 

iO  O 

l>-  TH 

0  00 

O  (M 


OO 
CO 
(N 


CO 
O 
CO 


OO  O  CO 

10 
iO 


>O 


<N  (M 


(M 


rH     t^     O     (M 

O    O    O    CO 

-HHCOOI^OOOCOrH 

OOiOC^^C^t^^ 

OOrHrHrH(MrH(N 


CO 

(M 


<N 


CO 


Tj<     O 


CO 


O  >O 


O 

& 


l> 

CO 
CO 


d  3    • 
I       a 


PQ 


90  DISCUSSION    OF   DEFLECTIONS    OF    THE    TWO    SLABS 

It  will  be  seen  from  Tables  1  and  3,  that  the  first  three  loads 
were  practically  the  same  for  both  slabs.  In  the  Norcross  slab 
load  3,  of  18  tons,  stressed  the  steel  up  to  the  yield  point,  but  in 
the  mushroom  slab  the  stress  was  so  small,  (being  in  fact  less  than 
ten  per  cent  of  the  former)  as  probably  not  to  remove  all  the  com- 
pression from  the  concrete  in  which  it  was  embedded.  Indeed  the 
load  on  the  latter  slab  became  five  times  as  much,  90  tons,  before 
its  steel  approached  the  yield  point,  at  which  time  it  was  carrying 
about  twice  the  load  which  caused  the  complete  failure  of  the 
Norcross  slab. 

Moreover  the  deflection  of  the  Norcross  slab  under  load 
3,  was  twenty-two  times  that  of  the  mushroom  slab  under  the 
same  load.  This  result  is  in  full  accord  with  slab  theory  which  shows 
that  the  central  deflection  of  a  continuous  diagonal  beam  with  fixed 
ends  uniformly  loaded  with  one  sixth  of  the  total  load  on  the  slab 
and  having  the  same  thickness  and  reinforcement  as  the  diagonal 
belt,  would  have  more  than  six  times  the  central  deflection  of  the 
slab,  while  the  stress  in  its  steel  would  be  three  or  four  times  as 
much.  This  gives  a  measure  of  the  effect  of  slab  action. 

By  the  phrase  "slab  action"  we  designate  the  increased  strength 
and  stiffness  of  the  slab  by  reason  of  its  resistance  to  circumferential 
stresses  around  the  columns  and  around  the  center  of  the  panel. 

Furthermore,  if  this  continuous  beam  be  compared  with  a  simple 
beam  uniformly  loaded  and  having  the  same  reinforcement,  the 
latter  would  have  five  times  the  deflection  of  the  continuous  beam, 
or  thirty  times  that  of  the  slab,  while  the  stress  in  the  steel  would 
be  one  and  one-half  times  that  in  the  continuous  beam,  and  six  or 
seven  times  that  in  the  slab.  This  last  exhibits  the  effect  of  canti- 
lever action  combined  with  slab  action. 

The  apparent  discrepancy  between  the  observed  ratio  of  de- 
flections in  these  two  slabs  of  22  and  the  just  computed  deflections 
of  30,  is  to  be  accounted  for  by  the  fact  that  the  computation 
assumed  equal  spans,  whereas  the  Norcross  span  was  assumed 
to  be  diminished  from  144"  to  132"  by  the  column  plate.  A  re- 
duction of  the  span  of  this  amount  will  change  the  computed  de- 
flections in  the  ratio  of  1443  :  1323  :  :  30  :  23  which  is  in  practical 
agreement  with  the  observed  result  of  22. 


SUMMARY   OF   TEST   OF   MUSHROOM   SLAB  91 

By  the  phrase  "cantilever  action"  we  designate  the  increased 
strength  and  stiffness  which  is  due  to  the  continuity  of  the  beam 
or  slab  at  its  supports  so  that  it  is  convex  upwards  at  such  points. 

While  the  concentration  of  the  loading  toward  the  middle  of 
the  panel,  such  as  was  the  case  in  this  test,  may  prevent  any  pre- 
cise agreement  of  these  numerical  estimates  based  on  uniform 
loading  with  the  results  of  the  tests,  they  cause  the  general  agree- 
ment shown  in  the  tables  and  tend  strongly  to  sustain  our  confi- 
dence in  the  validity  of  the  analysis  from  which  these  concordant 
approximate  estimates  are  obtained. 

The  amazing  difference  in  the  strength  and  stiffness  of  these 
two  slabs,  which  contain  practically  the  same  amount  of  concrete 
and  steel,  is  due  to  the  difference  of  principle  of  their  construction, 
which  may  be  summarized  for  the  mushroom  system  by  consider- 
ing its  slab  action  and  its  cantilever  action  under  the  following 
counts,  viz: 

1st.  Circumferential  slab  stresses  are  most  economically  and 
effectively  provided  for  by  the  ring  rods  around  the  column  heads. 

2nd.  The  size  of  the  mushroom  heads  is  such  as  to  make  the 
belts  so  wide  as  to  provide  reinforcement  over  the  entire  area  of 
the  slab,  thus  securing  slab  action  in  the  central  part  of  the  panel 
where  the  belts  lie  near  the  lower  surface. 

3rd.  The  reinforcing  belts  cover  a  wide  zone  at  the  top  of 
the  slab  over  the  columns  and  mushroom  head,  which  thus  provides 
resistance  to  tension,  and  ensures  effective  cantilever  and  slab  action. 

4th.  Concrete  is  thus  stressed  in  compression  at  the  bottom 
of  the  slab  for  a  wide  zone  around  the  columns. 

5th.  Under  a  load  equal  to  the  breaking  load  of  the  Norcross 
slab,  amounting  to  thirty-eight  tons,  the  mushroom  slab  deflected 
at  first  only  1/8",  but  after  exposure  to  rain  and  great  changes  of 
temperature  for  seven  days  had  somewhat  softened  the  concrete 
the  deflection  increased  to  1/4". 

6th.  The  first  crack  appeared  underneath  the  edge  of  the 
slab  across  the  side  belt  under  load  No.  5,  of  fifty-six  tons,  with  a 
center  deflection  of  0.4"  and  an  average  deflection  at  the  middle 
of  side  belts  of  0.25". 

7th.  No  cracks  appeared  on  the  upper  side  of  slab  at  the 
edge,  nor  were  any  seen  elsewhere,  until  load  No.  7,  of  90  tons  was 
applied,  when  the  yield  point  of  the  steel  was  evidently  nearly  or 
quite  reached,  giving  a  center  deflection  of  1/2". 


92 


FAILURE    OF    MUSHROOM    SLAB 


Fig.  18.     Failure  of  Mushroom  Slab. 


Fig.  19.     Failure  of  Mushroom  Slab.     Load  Removed. 


FAILURE    OF    MUSHROOM   TEST    SLAB  93 

8th.  The  slab  carried  its  final  load  of  over  120  tons  for  twenty- 
four  hours  without  giving  way.  It  demonstrated  the  impossibility 
of  its  sudden  failure  by  gradually  yielding  until  it  reached  a  final 
deflection  of  some  nine  inches,  as  seen  in  the  views  of  Dec.  17th 
and  24th,  Figs.  18  and  19. 

9th.  While  the  slab  steel  in  each  belt  was  the  same  as  in  the 
Norcross  slab,  the  crossing  of  the  belts  increased  the  percentage 
of  slab  reinforcement  so  much  above  that  of  the  simple  belt  rein- 
forcement that  stress  in  the  steel  did  not  pass  the  yield  point  and 
the  failure  was  largely  due  to  the  giving  way  of  the  concrete  around 
the  cap,  but  partly  to  some  yielding  at  the  line  of  weakest  ultimate 
resistance,  both  of  which  statements  are  confirmed  by  the  view  of 
Dec.  24th,  Fig.  19,  where  the  removal  of  the  loading  permits  the 
irregular  circular  line  previously  mentioned  to  be  made  out  at  a 
distance  from  the  center  of  each  column  of  somewhat  less  than  L/2. 

Less  steel  is  required  in  this  system  than  in  the  Nor- 
cross slab  for  the  same  limiting  stresses.  Since  the  steel  in  this 
slab  did  not  pass  the  yield  point  any  greater  percentage  of  reinforce- 
ment would  be  useless  and  would  not  increase  the  strength  of  the 
slab.  It  has  been  found  that  good  practice  requires  a  percentage 
of  steel  dependent  in  the  following  manner  upon  the  thickness 
of  the  slab: 

If  d  =  L/35  the  belt  reinforcement  =  0.2% 
lid  =  L/24  the  belt  reinforcement  =  0.3% 
lid  =  L/20  the  belt  reinforcement  =  0.4% 

Comparision  of  the  steel  in  the  test  slabs:  Norcross.  Mushroom. 

Size  of  slab 16'  x  16'  18.4'  x   17.8' 

Area  of  slab 256  sq.  ft.  328  sq.  ft. 

Length  of  3/8"  rods  in  the  slab 1188  ft.  1450  ft. 

Weight  of  3/8"  rods  in  the  slab 446  Ibs.  544  Ibs. 

Weight  of  Plates  or  Heads  in  the  slab. .  .  268  Ibs.  435  Ibs. 

Total  weight  of  steel  in  the  slab 714  Ibs.  979  Ibs. 

Weight  of  steel  per  square  foot  of  slab..  2. 8  Ibs.  3  Ibs. 

Area  of  Panel  12  x  12  ft. 144  sq.  ft.  144  sq.  ft. 

Length  of  slab  rods  per  panel 638  ft.  638  ft. 

Weight  of  slab  rods  per  panel 239  Ibs.  239  Ibs. 

Weight  in  plates  or  heads  per  panel 67  Ibs.  109  Ibs. 

Total  weight  of  steel  per  panel 306  Ibs.  348  Ibs. 

Weight  of  steel  per  square  foot  of  panel.  2  1/8  Ibs.  2  5/12  Ibs. 


SUGGESTIONS  REGARDING  THE  CONSTRUCTION 

AND  FINISH  OF  FLOOR  SLABS 

By  C.  A.  P.  TURNER 

18.  THE  EXECUTION  OF  WORK:  Construction  work  of  any  kind 
involves  a  great  responsibility,  not  only  on  the  part  of  the  designer, 
but  also  on  the  part  of  those  in  charge  of  the  work,  and  that  re- 
sponsibility is  for  the  safety  of  those  erecting  the  work. 

Perhaps  the  construction  of  no  type  of  building  is  so  free  from 
hazard  and  risk  to  the  lives  of  those  erecting  it  as  reinforced  con- 
crete construction  when  scientifically  designed  and  intelligently 
executed. 

During  the  last  ten  or  twelve  years,  the  manufacturers  of  Port- 
land Cement,  have  through  improvements  in  methods  of  manu- 
facture and  great  reduction  in  cost,  placed  this  material  on  the 
market  at  such  reasonable  rates  that  it  has  given  a  remarkable 
impetus  to  the  construction  of  concrete  work  in  all  lines.  Since,  as  a 
material  of  construction,  it  has  but  recently  come  into  general  use, 
it  is  not  surprising  that  a  large  part  of  the  engineering  and  archi- 
tectural profession  have  not  yet  become  so  familiar  with  its  char- 
acteristics, but  that  designs  lacking  in  conservatism  from  a  scientific 
standpoint  have  been  frequently  made,  and  this  combined  with 
the  execution  of  the  work  by  unskilled  contractors,  has  resulted  in  a 
number  of  instances  in  needless  sacrifice  of  life  and  large  property 
losses,  such  as  a  more  thorough  knowledge  and  study  of  the  char- 
acteristics of  the  material  should  entirely  prevent. 

It  would  be  neglect  of  duty  to  fail  even  in  this  short  discussion 
to  call  attention  pointedly  to  those  properties  and  characteristics 
of  concrete  which  must  be  known  and  appreciated  by  the  engineer 
and  constructor  in  order  that  he  may  avoid  the  serious  disasters  into 
which  those  ignorant  or  forgetful  of  them  have  been  too  frequently 
led. 

THE  HARDENING  OF  CONCRETE:  Concrete  may  be  defined  as 
an  artificial  conglomerate  stone  in  which  the  coarse  aggregate  or 
space-filler  is  held  together  by  the  cement  matrix.  The  cement 
should  conform  to  the  Standard  Specifications  for  Cement,  recom- 
mended by  the  American  Society  for  Testing  Materials. 


HARDENING   OF   CONCRETE  95 

The  contractor  and  architect  should,  at  least,  see  to  it  that  the 
cement  is  finely  ground,  and  that  it  meets  the  requirements  of  the 
boiling  test.  This  last  may  be  readily  made  by  forming  pats 
of  the  cement  of  3J  to  4  inches  in  diameter  on  a  piece  of  glass,  knead- 
ing them  thoroughly  with  just  enough  moisture  to  make  them  plastic, 
so  that  they  will  hold  their  shape  without  flowing,  and  taper  to  a 
thin  edge.  Store  the  pats  under  a  moist  cloth  at  a  temperature  of 
sixty-five  to  seventy-five  degrees  Fahr.  for  a  period  of  24  hours. 
Then  place  the  pats  in  a  kettle  or  pan  of  cold  water,  and  after  raising 
the  temperature  of  the  water  to  the  boiling  point,  continue  boiling 
for  a  period  of  four  hours.  If  the  pats  do  not  then  show  cracks, 
and  if  they  harden  without  cracking  or  disintegrating,  the  con- 
structor may  be  satisfied  that  the  cement  is  suitable  for  use  in  the 
work.  Coarse  grinding  reduces  the  sand-carrying  capacity  of  the 
cement,  and  its  consequent  efficiency. 

The  function  assigned  to  the  concrete  element  in  the  combina- 
tion of  reinforced  concrete  is  to  resist  compressive  stresses  in  bend- 
ing; but  when  first  mixed  the  concrete  is  nothing  more  than  mud, 
and  in  order  for  it  to  become  the  hard,  rigid  material  necessary  to 
fulfill  its  function  in  the  finished  work  it  must  evidently  pass  in  the 
process  of  hardening  thru  all  stages  and  varying  degrees  of  hardness 
from  mud  and  partly  cured  cement  to  the  final  stage  of  hard,  rigid 
material.  This  curing  or  hardening  being  a  chemical  process,  does 
not  occur  in  any  fixed  period  of  time,  save  and  except  the  temper- 
ature conditions  are  absolutely  constant.  Hence  the  time  at  which 
forms  may  be  safely  removed  is  not  to  be  reckoned  by  a  given  number 
of  days,  but  rather  it  must  be  determined  by  the  degree  of  hardness 
attained  by  the  cement.  In  other  words,  during  warm  summer 
weather,  concrete  may  become  reasonably  well  cured  in  twelve  or 
fifteen  days.  If  the  weather,  however,  is  rainy  and  chilly,  it  may 
not  become  cured  in  a  month.  In  the  cold,  frosty  weather  of  the 
spring  and  autumn,  unless  warm  water  is  used  in  the  mix,  the  con- 
crete may  require  two  or  three  months  to  become  thoroughly  cured, 
while  by  heating  the  mixing  water,  whenever  the  temperature  is 
below  50  degrees  Fahr.,  the  concrete  will  harden  approximately  as 
it  does  during  the  more  favorable  season. 

Concrete  which  has  been  chilled  by  the  use  of  ice  cold  water, 
or  that  has  become  chilled  within  the  first  day  or  two  of  the  time  it 
is  cast,  has  this  peculiarity,  that  it  is  very  difficult  indeed  for  the 
most  expert  to  determine  when  it  is  in  such  condition  that  it  will 
retain  its  shape  after  the  removal  of  the  forms.  Once  having  been 
chilled  in  the  early  stages,  it  goes  through  consecutive  stages  of 


96  POURING   CONCRETE 

sweating  with  temperature  changes,  and  during  these  periods  it 
sometimes  happens  that  the  concrete  diminishes  in  compressive 
strength,  and  if  the  props  are  removed  it  sags  and  gets  out  of  shape. 
Such  deformation  will  generally  result  in  checks  and  fine  cracks, 
though  there  may  not  be  any  serious  diminition  of  the  ultimate 
strength.  These  checks  may  be  prevented  as  explained  above  by 
the  simple  method  of  heating  the  mixing  water  whenever  the  tem- 
perature has  dropped  below  50  degrees  Fahr.  In  colder  weather, 
that  is  below  the  freezing  point,  not  only  must  the  water  be  heated, 
but  as  a  rule  the  sand  and  stone  too,  also  a  little  salt  may  be  ad- 
vantageously used.  The  work  must  then  be  properly  housed  and 
kept  warm  for  at  least  three  weeks  subsequent  to  pouring. 

USE  OF  SALT  IN  COLD  WEATHER  :  We  have  mentioned  the  use  of 
salt  in  cold  weather.  The  action  of  salt  is  two-fold:  It  retards  the 
setting  and  thus  enables  us  to  use  water  heated  to  a  higher  temper- 
ature than  we  could  use  without  salt.  It  also  lowers  the  freezing 
point.  Should  the  concrete  then  be  frozen  at  the  subsequent  sweat- 
ing period  which  occurs  with  a  rise  in  temperature,  the  salt  retains 
the  necessary  moisture  for  crystallization  because  of  its  affinity  for 
moisture,  thus  preventing  the  softened  concrete  from  drying  out 
and  disintegrating  through  lack  of  moisture  to  enable  it  to  crystalize 
and  harden  properly.  The  amount  of  salt  to  be  used  is  about  a 
cup  to  the  sack  of  cement  with  the  temperature  from  18  to  20  de- 
grees Fahr.  If  the  temperature  is  below  this,  increase  the  amount 
of  salt,  and  when  working  below  zero  Fahr.,  use  not  less  than  two 
cups  of  salt  to  the  bag  of  cement. 

POURING  CONCRETE:  Bad  work  frequently  results  from  im- 
proper pouring,  or  casting  of  the  work.  In  filling  the  forms,  the 
lowest  portion  of  the  forms  should  be  filled  first.  A  column  should 
be  filled  from  the  center  and  not  from  the  side  of  the  cap.  Filling 
from  the  center  will  insure  a  clean  smooth  face  when  the  forms  are 
removed.  Filling  from  the  side  will  frequently  give  a  bad  surface 
because  the  mortar  will  flow  into  the  center  of  the  column  through 
the  hooping,  leaving  the  coarse  aggregate  with  voids  unfilled  at  the 
outside.  As  more  concrete  is  then  poured  in,  the  voids  between  the 
core  and  the  out  side  portion  will  become  filled,  and  the  soft  mor- 
tar will  not  be  able  to  flow  back  to  completely  fill  the  voids  between 
the  hooping  and  the  casing.  Where  the  spacing  of  the  hooping  is 
wide,  this  is  not  so  important,  but  it  becomes  very  important  where 
the  spiral  used  has  close  spacing.  It  is  better  to  cast  the  column  and 
mushroom  frame  complete,  continuing  to  pour  the  concrete  over 
the  center  of  the  column  so  that  it  always  flows  from  the  column 


TEST    FOR    HARDNESS.       LAP  97 

into  the  Mushroom  slab  rather  than  the  reverse.  All  splices  must 
be  made  in  a  vertical  plane,  in  a  beam  preferably  at  the  middle  of 
the  span,  and  in  a  slab  at  a  center  line  of  a  panel. 

TEST  FOR  HARDNESS  IN  WARM  WEATHER:  We  have  pointed 
out  that  the  criterion  governing  the  safe  removal  of  forms  is  the 
hardness  or  rigidity  of  the  concrete.  A  test  of  hardness  in  concrete 
not  frozen  may  be  made  by  driving  a  common  eight-penny  nail 
into  it;  the  nail  should  double  up  before  penetrating  more  than 
half  an  inch.  The  concrete  should  further  be  hard  enough  to 
break  like  stone  in  knocking  off  a  piece  with  the  hammer. 
Noting  the  indentation  under  a  blow  with  the  hammer,  gives  a 
fair  idea  of  its  condition  to  those  having  experience. 

Subcentering,  as  provided  in  the  appended  specification,  is  a 
desirable  method  of  preventing  deformation,  where  the  use  of  the 
forms  is  desired  for  upper  stories  before  the  concrete  is  fully  cured. 

TEST  FOR  HARDNESS  IN  COLD  WEATHER:  Concrete  freshly 
mixed  and  frozen  hard  will  not  only  sustain  itself  but  carry  a  large 
load  in  addition,  until  it  thaws  out  and  softens,  when  collapse  in 
whole  or  in  part  is  inevitable.  Partly  cured  concrete  if  frozen, 
sweats  and  softens  with  a  rise  in  temperature,  hence  in  cold  weather 
there  is  danger  of  mistaking  partly  cured  concrete  made  rigid  by 
frost  for  thoroughly  cured  material.  In  fact  the  only  test  that  can 
be  depended  upon  with  certainty  in  cold,  frosty  weather,  is  to  dig  out 
a  piece  of  concrete,  place  a  sample  on  a  stove  or  hot  radiator,  and 
note  whether,  as  the  frost  is  thawed  out  of  it,  it  sweats  and  softens. 
This  gives  the  builder  and  engineer  a  perfectly  conclusive  test  of  the 
condition  of  the  concrete  as  to  whether  it  is  cured  or  merely  stiffened 
up  by  frost. 

LAP  OF  REINFORCEMENT  OVER  SUPPORTS:  Thoroughly  tying 
the  work  together  by  ample  lap  in  the  reinforcement  is  a  prime 
requisite  for  safety  in  any  form  or  type  of  construction.  This 
general  precaution  insures  toughness,  and  prevents  instantaneous 
collapse,  should  the  workman  exercise  bad  judgment  in  premature 
removal  of  forms. 

RESPONSIBILITY  OF  THE  ENGINEER:  The  steps  which  it  is 
possible  for  the  engineer  to  take  in  securing  safe  construction  are 
limited  in  the  first  place  to  the  production  of  a  conservative  design, 
and  one  which  will  present  toughness,  so  that  its  failure  under  over- 
load or  under  premature  removal  of  the  forms  will  be  slow  and 
gradual.  This  he  can  do,  and  this  we  believe  he  is  morally  bound 
to  do.  On  the  other  hand,  he  cannot  design  reinforced  concrete 


CRACKS   IN    CONCRETE 


work  which  will  hold  its  shape  without  permanent  deformation,  un- 
less it  is  properly  supported  until  the  concrete  has  had  time  under 
proper  conditions  to  become  thoroughly  cured. 

Concrete  in  setting  shrinks,  and  sometimes  cracks  by  reason  of 
this  shrinkage,  particularly  when  it  hardens  rapidly,  as  it  does  in 
hot  weather.  This  shrinkage  sets  up  certain  stresses  in  the  concrete, 
which,  combined  with  temperature  changes,  occasionally  manifest 
themselves  by  subsequent  cracks  in  the  work.  Such  checks  or 
cracks  do  not  of  necessity  indicate  weakness,  providing  the  concrete 
is  hard  and  rigid,  since  the  steel  is  intended  to  take  the  tensile 
stresses  and  the  concrete  the  compressive.  Such  checks  sometimes 
cause  an  unwarranted  lack  of  confidence  in  the  safety  and  stability 
of  the  work  arising  from  the  common  lack  of  familiarity  with  the 
characteristics  of  the  material.  For  example,  the  owner  of  a  frame 
building  would  never  imagine  it  to  be  unsafe  because  he  found  a  few 
season  checks  in  the  timber.  He  is  sufficiently  familiar  with  the 
seasoning  of  timber  to  understand  how  these  checks  occur,  and  that 
in  most  instances  they  do  not  mean  a  loss  of  strength,  since,  as  the 
timber  hardens  by  thoroughly  drying  out,  it  becomes  stronger,  as  a 
rule,  to  an  amount  in  excess  of  any  slight  weakness  which  might  be 
developed  by  ordinary  season  cracks  or  checks.  So  in  concrete, 
when  the  general  public  becomes  more  familiar  with  its  character- 
istics they  will  regard  as  far  less  important  than  they  now  do,  checks 
which  are  produced  by  temperature  and  shrinkage  stresses,  or 
possibly  by  slight  unequal  settlement  of  supports. 

PROPER  AND  IMPROPER  METHODS  OF  FLOOR  FINISH:  In  con- 
crete work  there  are  a  number  of  small  defects  which  occur  through 
failure  to  properly  manipulate  the  material,  for  which  the  designer 
of  the  engineering  part  of  the  work  is  frequently  censured  improperly. 
For  example,  cases  have  occurred  where  a  good  splice  was  not 
secured  owing  to  the  fact  that  in  very  hot  weather  the  stone  aggre- 
gate became  heated  in  the  sun  and  was  not  properly  cooled  down 
before  mixing  the  concrete,  and  so  the  water  dried  out  too  quickly, 
while  the  heat  in  the  stone  caused  the  cement  to  set  so  rapidly  that 
a  good  splice  to  the  previous  work  could  not  be  made. 

The  worst  trouble,  however,  which  has  been  observed,  is  that 
resulting  from  poor  surface  finish  of  floors.  Improper  methods  in 
common  practice  are  of  two  different  kinds.  One  is  the  attempt 
to  finish  the  work  approximately  at  the  time  it  is  cast,  making  the 
surface  finish  integral  with  the  slab.  The  difficulty  with  this  method 
of  finishing  lies  in  the  fact  that  as  soon  as  the  columns  are  cast  in 
the  story  above,  unequal  moisture  conditions  are  produced  around 


FLOOR    FINISH  99 

the  foot  of  the  column  owing  to  the  excess  of  moisture  in  the  column ; 
thus  the  concrete  in  the  surface  of  the  slab  around  and  near  the  foot 
of  the  column  is  expanded  by  the  excess  moisture,  and  it  ultimately 
shrinks,  and  leaves  a  series  of  spider  web  cracks  as  it  dries  out. 
This  will  occur  to  a  greater  or  less  extent  depending  on  the  humidity 
of  the  surrounding  atmosphere  during  the  curing  or  drying  out  of  the 
floor.  If  the  weather  is  dry  these  checks  will  be  very  pronounced 
indeed,  though  they  will  not  be  very  deep.  If  it  is  rainy  and  damp, 
and  the  floor  is  kept  soaked  all  the  time,  they  may  be  nearly  or  quite 
lacking. 

Another  objection  to  this  method  of  finish  is  that  unusual  pre- 
cautions must  be  taken  to  protect  the  floor  before  the  centering  can 
be  placed  for  a  story  above,  and  regardless  of  the  method  used  to 
protect  it  the  floor  usually  becames  scarred  and  deeply  scratched  before 
the  work  is  complete,  leaving  a  surface  difficult  to  satisfactorily  repair. 

Another  method  which  leads  to  bad  results  is  the  following: 
The  rough  slab  is  cast,  and  the  centering  removed  in  due  time,  the 
slab  cleaned  and  the  finish  coat  applied  in  a  sloppy  or  plastic  form, 
flowed  in  place,  screeded  to  approximate  surface,  and  then  allowed 
to  partly  set,  so  that  the  finishers  can  get  on  the  floor  and  trowel  it 
down.  A  floor  finished  in  this  manner  looks  well  when  the  work  is 
new.  It  does  not  wear  well  but  dusts  badly,  pits  and  rapidly  grows 
rough  and  ragged  under  trucking. 

The  correct  method  of  applying  floor  finish  is  as  follows: 
The  finish  coat  should  be  not  less  than  1  and  1/4  inches  to  1  and  1/2 
inches  in  thickness.  It  should  be  applied  after  the  rough  slab  has  been 
fairly  well  cured.  The  surface  of  the  rough  slab  should  be  thoroughly 
cleaned  of  dirt  and  laitance  and  thoroughly  soaked  with  water. 
Then  the  floor  finish,  a  mixture  preferably  of  one  part  of  cement  to 
one  and  one-half  sand  (the  sand  a  silicious  sand  with  grains  from 
1/8  inch  down,  if  such  can  be  secured),  should  be  thoroughly  mixed 
with  just  enough  water  to  make  an  extremely  stiff  paste,  one  which 
will  hold  its  form  if  squeezed  in  the  hand,  but  one  which  will  not  run 
or  flow,  and  will  need  a  fair  amount  of  tamping  to  bring  the  moisture 
to  the  surface.  This  concrete,  so  mixed,  should  be  applied  to  the 
rough  slab  in  blocks  of  from  four  to  five  feet  square,  first  grouting 
the  rough  slab  with  a  neat  cement  grout,  then  tamp  until  the  moist- 
ure is  brought  to  the  surface,  level  up  and  trowel  immediately. 
The  cement  finish  should  not  be  mixed  more  rapidly  than  it  can  be 
applied,  so  that  the  cement  will  not  be  killed  by  taking  a  partial  set 
before  troweling,  which  is  what  occurs  where  the  finish  is  applied 
sloppy,  and  the  workmen  wait  for  it  to  partly  harden  before  they  can 


100  STRIP    FILL    FLOORS 

get  on  it  to  trowel.  A  finish  applied  as  just  stated  will  stand  severe 
usage  and  last  for  several  years  without  showing  appreciable  evi- 
dence of  pitting,  dusting,  or  undue  wear. 

The  addition  of  ground  iron  ore,  to  the  amount  of  twenty 
pounds  to  the  barrel  of  cement,  appears  to  improve  the  finish  and 
give  it  a  more  pleasing  color. 

Checks  in  cement  finish  have  no  relation  whatever,  as  a  rule, 
to  the  strength  of  the  work.  They  will  invariably  occur  in  the 
cement  finish  where  the  finish  coat  is  too  thin.  When  it  is  less  than 
1  inch  or  3/4  inch  at  one  part  of  the  floor  with  1  1/4  inches  or  1  1/2 
inches  at  another,  the  surface  will  invariably  check  and  crack  badly  if 
applied  at  a  sloppy  consistency  and  allowed  to  partly  cure  before  it 
is  polished  down.  We  know  of  no  type  of  construction  where  there 
has  not  been  much  trouble  with  finished  surfaces  in  such  buildings 
as  have  come  under  our  observation.  But  experience  has  shown  us 
that  these  troubles  are  needless,  and  can  be  avoided  by  the  proper 
handling  and  application  of  the  finishing  coat. 

It  is  difficult  indeed  to  re-educate  those  who  profess  to  be 
cement  finishers,  whose  experience  has  been  largely  in  sidewalk 
finish,  or  work  of  that  character,  to  appreciate  the  necessity  for  a 
different  method  of  executing  work  in  a  building;  but  when  this  has 
been  accomplished  the  owner  will  have  the  use  of  a  floor  finish  free 
from  the  unpleasant  defects  above  pointed  out. 

STRIPS  AND  STRIP  FILL  FOR  WOOD  FLOORS:  The  proper  time 
for  the  application  of  the  strips  and  fill  is  immediately  after  the 
rough  slab  has  become  sufficiently  hardened  to  work  upon  it,  for  the 
reason  that  at  this  time  the  strips  may  be  spiked  to  the  partially 
hardened  concrete  and  wedged  up  or  lined  up  to  the  desired  level 
without  difficulty.  Then  the  strip  fill  can  be  put  in  with  the  same 
rig  that  is  used  to  cast  the  floor  slab. 

The  writer  prefers  the  strip  fill  of  the  same  mixture  as  the  slab 
except  where  the  loads  are  so  light  that  increased  strength  and 
stiffness  are  of  no  importance.  Then  a  one  to  three  and  one-half, 
four,  or  even  five,  mix  will  answer  the  purpose.  No  natural  cement  or 
lime  should  be  used  in  the  mixture,  since  when  it  is  used,  trouble 
almost  invariably  follows,  caused  by  its  extremely  slow  hardening 
and  its  retention  of  moisture  until  hardening  takes  place.  This 
moisture  frequently  swells  and  expands  the  flooring  to  such  an 
extent  that  it  springs  away  from  the  fastenings,  thereby  necessi- 
tating the  entire  relaying  of  the  floors.  Conservative  practice  ac- 
cordingly is  to  use  Portland  Cement  alone,  which  will  dry  out  far 
quicker  than  any  natural  cement  or  brown  lime. 


APPENDIX 


STANDARD  SPECIFICATION  FOR  REINFORCED 
CONCRETE   FLOORS 

By  C.  A.  P.  TURNER,  Consulting  Engineer 
Minneapolis,  Minn. 


Reinforcement.  Reinforcement  shall  be  of  sizes  of  bars  shown  on  the  accom- 
panying plans  and  details  which  form  a  part  of  this  specification. 

All  reinforcing  metal  shall  be  of  medium  open  hearth  or  Bessemer  steel, 
meeting  the  requirements  of  the  Manufacturers'  Standard  Specifications,  in 
composition,  ultimate  strength,  ductility  and  elastic  limit,  and  the  required 
bending  basis.  Hard  grade  may  be  used  for  slab  rods  only. 

Bending.  Bending  shall  preferably  be  done  cold.  If  the  column  rods  are 
heated  and  blacksmith  work  is  done,  care  must  be  exercised  that  the  steel  is  not 
burned  in  the  operation,  otherwise  it  will  be  condemned  by  the  engineer. 

Cement.  Cement  shall  be  of  good  quality  of  Portland  Cement,  of  a  brand 
which  has  been  upon  the  market  and  successfully  used  for  at  least  four  years, 
meeting  the  requirements  of  the  specification  adopted  by  the  American  Society 
for  Testing  Materials. 

The  contractor  shall  give  the  owner  the  opportunity  to  test  all  cement  de- 
livered, and  shall  furnish  the  use  of  testing  machine  for  this  purpose. 

The  cement  shall  be  delivered  in  good  condition  and  properly  protected 
under  suitable  cover  after  delivery  on  the  premises  so  that  it  may  not  be  damaged 
by  moisture. 

Sand.  Sand  used  in  the  concrete  work  shall  be  clean  and  coarse,  meeting 
the  requirements  and  approval  of  the  engineer  and  architect. 

Stone.  Stone  used  shall  be  sound,  hard  stone,  free  from  lumps  of  clay  and 
other  soft  unsatisfactory  material,  or  hard  smelter  slag  may  be  used.  In  size  it 
shall  be  crushed  to  pass  a  1-inch  ring,  for  slabs  and  columns,  and  shall  be  screened 
free  from  dirt  and  dust. 

Concrete.  All  concrete  shall  be  mixed  in  a  standard  batch  machine  to  the 
consistency  of  brick  mortar,  so  that  it  will  flow  slowly  and  require  only  puddling 
around  the  reinforcement. 

Concrete  shall  be  thoroughly  mixed  in  the  following  proportions:  one  part 
cement,  meeting  the  requirements  of  the  standard  specifications;  two  parts 
clean,  coarse  sand  free  from  clay,  loam  or  other  impurities;  and  four  parts  crushed 
stone  or  clean  gravel. 

The  concrete  shall  be  poured  in  the  low  portions  of  the  forms  first.  That  is, 
it  shall  be  poured  directly  into  the  column  boxes,  beam  boxes,  etc.,  before  it  is 


,102*  • 


STANDARD    SPECIFICATIONS 


poured  on  the  slab.  It  shall  be  so  placed  that  it  will  be  forced  to  flow  as  little 
as  possible  to  get  to  the  required  position,  since  by  flowing,  the  cement  is  readily 
separated  from  the  mixture. 

Splices.  Splices  in  beams  or  slabs  are  to  be  made  in  a  vertical  plane,  prefer- 
ably in  the  center  of  the  panel  or  beam. 

Proportions.  Each  sack  of  cement  shall  be  considered  equivalent  to  one 
cubic  foot  in  volume,  and  the  mixture  of  the  cement,  sand  and  stone  used  in  the 
concrete  shall  be  proportioned  by  volume  on  this  basis  and  as  hereinafter  specified. 

Concrete  for  footings,  columns,  beams  and  rough  slabs  throughout  shall 
consist  of  a  mixture  of  one  cement,  two  sand  and  four  of  crushed  stone. 

For  the  retaining  walls,  the  concrete  mixture  shall  be  one  cement,  three  sand 
and  five  parts  of  stone. 

Concrete  in  which  the  cement  has  attained  its  initial  set  shall  not  be  used  on 
the  work.  Concrete,  however,  which  has  slopped  out  of  the  mixer,  if  cleaned 
up  within  a  short  time,  not  over  every  half  hour,  may  be  put  back  in  the  mixer, 
and  after  being  thoroughly  mixed  again  with  water  may  be  used  on  the  work. 

Forms.  All  forms  for  the  reinforced  concrete  shall  be  substantially  made 
and  true  to  line.  Any  irregularities  due  to  defective  workmanship  in  this  re- 
spect, shall  be  made  good  as  directed  by  the  architect,  by  dressing  down  the 
finished  work,  or  removal  and  properly  replacing  it  in  case  that  it  cannot  be 
satisfactorily  done. 

A  fair  quality  of  lumber,  preferably  1x6  square  edge  fencing  shall  be  used 
for  the  slab  forms.  This  lumber  shall  be  dressed  on  the  side  next  to  the  concrete 
except  where  plaster  is  specified  by  the  architect  for  office  finish,  in  which  case 
the  rough  side  of  the  boarding  shall  be  placed  upwards,  next  to  the  concrete. 

Column  Forms.  Column  forms  shall  be  made  up  with  plank  not  less  than 
1§  inches  thick  and  stayed  at  intervals  not  more  than  18  inches  vertically  be- 
tween bands  or  straps  and  shall  fit  closely  at  the  corner  joints,  or  the  forms  may 
be  made  of  sheet  metal. 

Removal  of  the  Forms.  Forms  shall  not  be  removed  under  the  most 
favorable  conditions,  prior  to  two  weeks'  time,  and  under  less  favorable  con- 
ditions where  the  temperature  is  lower  than  50  °  until  the  concrete  is  hard  and 
rigid. 

The  superintendent  will  keep  in  mind  the  fact  that  it  is  not  the  number  of 
days  time  which  has  elapsed  since  placing  the  concrete  which  shall  determine  the 
earliest  removal  of  the  forms,  but  rather  how  rapidly  the  concrete  has  thoroughly 
cured  and  hardened  and  that  the  concrete  may  be  readily  stiffened  up  by  cold 
and  frost  which,  when  it  thaws,  will  sweat  and  fail  to  maintain  the  desired  form. 

Sub=Centering.  Where  a  series  of  floors  are  cast  one  above  the  other,  sub- 
centering  of  substantial  posts  about  10  feet  centers  shall  be  kept  in  place  until 
there  are  at  least  two  supporting  slabs  that  are  well  cured  and  hard  so  that  the 
concrete  may  not  be  overstained  in  the  early  stages  of  hardening. 

Placing  and  Inspection  of  Reinforcement.  BEFORE  COMMENCING  THE 
CONCRETE  WORK,  the  reinforcement  shall  be  properly  placed  and  inspected  by 
the  architect  or  the  engineer  representing  the  owner,  and  not  until  after  this 
inspection  and  approval  may  the  work  of  casting  the  floor  proceed. 

The  floor  slab  rods  shall  be  wired  together  to  hold  them  in  the  position  as 
shown  on  the  plans.  Special  attention  being  given  to  placing  the  rods  in  belts 
of  the  width  of  the  mushroom  frame  and  fairly  uniform  spacing,  although  this  is 


STANDARD    SPECIFICATIONS  103 

of  less  importance  than  keeping  to  the  general  distribution  through  the  full  width 
of  the  belts  of  reinforcement. 

In  placing  the  floor  slab  rods,  all  those  running  from  column  to  column 
directly  on  one  side  of  a  panel  shall  be  placed  first,  then  those  running  at  right 
angles,  next  all  those  in  one  diagonal  belt,  and  then  those  in  the  other  diagonal. 

Where  a  belt  of  slab  rods  runs  parallel  to  a  wall  place  one  rod  at  bottom  on 
forms.  Then  see  that  belts  normal  and  diagonally  are  placed,  following 
up  with  slab  rods  parallel  to  the  wall  on  the  top  of  normal  and  diagonal  belts. 

In  wiring  the  rods  together  it  is  desirable  to  use  No.  16  soft  annealed  wire, 
taking  a  piece,  say  a  yard  long,  fastening  an  intersection,  then  carry  the  wire 
diagonally  to  the  next  intersection,  taking  a  half  hitch  and  proceed  until  this 
piece  is  used  up  and  making  the  end  fast.  Then  star^  with  a  new  piece  and 
proceed  as  before. 

Two  lines  of  ties,  crossing  and  normal  to  the  intersecting  belts  at  the  center 
will  hold  these  rods  in  position  very  nicely. 

A  similar  tie  across  the  parallel  belts,  and  a  suitable  number  of  fastenings 
around  the  mushroom  head  are  required  to  hold  the  bars  in  position. 

Floor  Finish.  The  finish  coat  on  the  rough  slab  shall  not  be  less  than  1  inch 
thick,  and  the  rough  slab  shall  be  prepared  for  its  reception  as  follows : 

The  slab  shall  be  thoroughly  scrubbed  with  a  steel  brush  and  water,  and  then 
after  it  has  been  thoroughly  cleaned  from  dirt  and  laitance  it  shall  be  kept  wet 
for  at  least  six  hours.  The  surface  shall  then  be  coated  with  neat  cement  grout 
and  the  finish  coat  applied. 

The  finish  coat  shall  consist  of  a  mixture  of  one  cement  to  one  and  one-half 
clean,  coarse  sand.  The  finish  coat  shall  be  mixed  with  just  enough  water  to 
make  a  very  stiff  paste  and  not  enough  to  make  it  soft  and  sloppy.  It  shall  be 
tamped  in  place  and  troweled  to  a  smooth  finish. 

Mixing  the  material  wet  and  sloppy  renders  it  necessary  to  wait  until  the 
material  hardens  somewhat  before  it  is  possible  to  polish  it  down.  In  allowing 
it  to  partly  harden  the  finisher  is  then  obliged  to  break  up  the  surface  of  partly 
hardened  cement  which  results  in  a  finished  surface  that  will  dust  badly,  pit 
readily  and  wear  rough  under  subsequent  use,  so  that  this  method  should  not 
be  employed. 

This  finish  coat  is  to  be  blocked  off  in  squares  along  the  center  line  of 
columns,  and  joints  shall  be  made  in  this  coat  between  panel  joints  at  five  to 
six  foot  intervals. 

Conduits.  Before  casting  the  concrete,  the  concrete  contractor  shall  see 
that  the  electric  contractor  has  placed  the  necessary  conduits  for  the  wires.  These 
shall  be  kept  above  the  reinforcement  wherever  they  come  in  the  center  of  a 
panel,  the  idea  being  to  have  these  conduit  pipes  above  the  steel  and  dip  down 
into  the  socket  at  the  junction,  or  to  use  a  special  deep  socket  which  would  be 
prefered  by  the  engineer. 

These  conduit  pipes  should  be  carried  below  the  level  of  the  reinforce- 
ment around  the  mushroom  heads  where  the  reinforcement  is  of  necessity  near 
the  top  of  the  slab. 

Depositing  Concrete  in  Warm  Weather.  When  the  concrete  is  deposited 
in  temperatures  above  70°  Fahr.,  the  slab  shall  be  thoroughly  wet  down  twice 
a  day  for  two  days  after  it  has  been  cast.  Any  preliminary  shrinkage  cracks 
which  occur  on  the  surface  of  the  slab  due  to  too  rapid  drying  shall  be 
immediately  filled  with  liquid  cement  grout. 


104  STANDARD    SPECIFICATIONS 

Any  concrete  work  indicating  that  it  has  not  been  thoroughly  mixed  in  the 
required  proportions  shall  be  dug  out  and  replaced  as  directed  by  the  engineer 
and  architect. 

Placing  Concrete  in  Cold  Weather.  Where  the  temperature  is  below  45  ° 
Fahr.,  the  water  shall  be  heated  to  a  temperature  of  at  least  110  °.  Where  the 
temperature  is  below  30  °  Fahr.,  artifical  heat  shall  be  used  to  assist  in  curing 
the  concrete,  and  this  must  be  continued  until  such  a  time  as  the  slab  is  thorough- 
ly cured  and  dry  throughout. 

Pouring  Concrete.  In  the  mushroom  system  concrete  shall  be  poured 
over  the  center  of  the  column  until  the  column  is  filled.  Then  the  pouring 
shall  be  continued  until  the  mushroom  and  mushroom  frame  is  filled  up  so  that 
the  concrete  will  flow  from  the  column  toward  the  center  of  the  slab  and  not 
from  the  center  of  the  slab  toward  the  column.  In  this  way  solid  concrete 
without  joints  and  planes  of  imperfect  bond  will  be  secured  around  and  in  the 
vicinity  of  column  heads,  where  it  is  most  needed. 

Test.  No  test  shall  be  made  until  the  concrete  is  thoroughly  cured,  is  dry, 
hard  and  rigid  throughout.  Ninety  days  of  good  drying  weather  at  a  tempera- 
ture above  60  °  Fahr.,  either  natural  or  artificial,  shall  be  the  criterion  as  to  when 
the  test  of  double  the  working  capacity  can  be  reasonably  made. 

General.  It  is  the  general  intent  of  this  specification  to  require  first  class 
work  in  all  particulars,  and  work  unsatisfactory  to  the  engineer  and  architect 
representing  the  owners  shall  be  made  good  by  the  contractor  as  they  direct. 


PRINTED    BY 
HEYWOOD 

MINNEAPOLIS 


LIST  OF  ONE  HUNDRED  BUILDINGS  SELECTED  FROM 
MORE  THAN  A  THOUSAND  DESIGNED  ON  THE 

MUSHROOM  SYSTEM 

1906  Johnson-Bovey  Go's.  Bldg Minneapolis,  Minn. 

1906  Hoffman  Building Milwaukee,  Wis. 

1907  Bostwick  Braun  Bldg Toledo,  Ohio 

1907  Lindeke  Warner  Bldg St.  Paul,  Minn. 

1907  Hamm  Brewery  Bldg "       "       -« 

1907  Smythe  Building Wichita,  Kans. 

1907  Forman  Ford  Bldg Minneapolis,  Minn. 

1907  Grellet  Collins  Bldg Philadelphia,  Pa. 

1907  Parsons  Scoville  Bldg Evansville,  Ind. 

1907  Born  Building Chicago,  111. 

1908  South  Dakota  State  Capitol Pierre,  S.  D. 

1908  Merchants  Ice  &  Cold  Storage  Bldg Cincinnati,  Ohio 

1908  St.  Mary's  Hospital Kansas  City,  Mo. 

1908  John  Deere  Plow  Co Omaha,  Nebr. 

1908  Minn.  State  Prison  Bldgs  ...  (6) Stillwater,  Minn. 

1908  Ripley  Apartments Tacoma,  Wash. 

1908  Velie  Motor  Bldg Moline,  111. 

1908  Park  Grant  Morris  Bldg Fargo,  N.  D. 

1909  Con  P.  Curran  Bldg St.  Louis,  Mo. 

1909  Manchester  Biscuit  Go's.  Bldg Fargo,  N.  D. 

1909  Blue  Line  Transfer  &  Storage  Bldg Des  Moines,  la. 

1909  Cutler  Hardware  Bldg Waterloo,  la. 

1909  Mass.  Cotton  Mills Boston,  Mass. 

1909  McMillan  Packing  Co St.  Paul,  Minn. 

1909  Vancouver  Ice  and  Cold  Storage  Co Vancouver,  Bldg. 

1909  Omaha  Fireproof  Storage  Bldg Omaha,  Nebr. 

1909  J.  I.  Case  Bldg Oklahoma  City,  Okla. 

1909  Tibbs  Hutchings  &  Co Minneapolis,  Minn. 

1909  Snead  Mfg.  Bldg Louisville,  Ky. 

1909  New  England  Sanitary  Bakery  Bldg Decatur,  111. 

1909  International  Harvester  Bldg Milwaukee,  Wis. 

1909  Congress  Candy  Co Grand  Forks,  N.  D. 

1910  Y.  M.  C.  A.  Bldg Winnipeg,  Man. 

1910  West  Publishing  Go's.  Bldg St.  Paul,  Minn. 

1910  Beatrice  Creamery  Bldg Lincoln,  Nebr. 

1910  Iten  Biscuit  Co Omaha,  Nebr. 

1910  Turner  Moving  &  Storage  Bldg Denver,  Colo. 

1910  Congress  Realty  Go's.  Bldg Portland,  Me. 

1910  Sniders  &  Abrahams  Bldg Melbourne,  Australia 

1910  Strong  &  Warner  Bldg St.  Paul,  Minn. 

1910  Lexington  High  School  Bldg St.  Paul,  Minn. 

1910  Weicker  Transfer  &  Storage  Bldg Denver,  Colo. 

1910  Chehallis  County  Court  House Montesano,  Wash. 

1910  Missouri  Glass  Go's.  Bldg St.  Louis,  Mo. 

1910  Industrial  Bldg Newark,  N.  J. 

1910  Revel  &  Wagner  Bldg Little  Rock,  Ark. 

1910  Jobst  Bethard  Bldg Peoria,  111. 

1910  International  Harvester  (Keystone  Works)..  .Sterling,  111. 


1910  Patterson  Hotel Bismarck,  N.  D. 

1910  O'Neil  Bldg Akron,  Ohio 

1911  Lindsay  Bldg Winnipeg,  Man. 

1911  King  George  Hotel. Saskatoon,  Sask. 

1911  Northern  Cold  Storage  Bldg Duluth,  Minn. 

1911  Leighton  Supply  Co Fort  Dodge,  la. 

1911  Kinsey  Bldg Toledo,  Ohio 

1911  Lozier  Motor  Bldg Detroit,  Mich. 

1911  Mullin  Warehouse  Bldg Cedar  Rapids,  la. 

1911  Griggs  Cooper  &  Co St.  Paul,  Minn. 

1911  Swift  Canadian  Go's.  Bldgs Vancouver,  B.  C. 

1911  McKenzie  Bldg Brandon,  Man. 

1911  Swift  Canadian  Go's.  Bldg Fort  William,  Ont. 

1911  Commerce  Bldg St.  Paul,  Minn. 

1911  Experimental  Eng.  Bldg.  Univ.  of  Minn Minneapolis,  Minn. 

1911  St.  Paul  Bread  Go's.  Bldg St.  Paul,  Minn. 

1911  Rust  Parker  Martin  Bldg Duluth,  Minn. 

1912  Woodward  Wight  Co.  Ltd.  Bldg New  Orleans,  La. 

1912  Internationa]  Harvester  Go's.  Bldg Fort  William,  Ont. 

1912  H.  W.  Johns-Manville  Bldgs. ...  (3) Finderne,  N.  J. 

1912  Cooledge  Bldg Atlanta,  Ga. 

1912  Lawrence  Leather  Go's.  Bldg .  .  .Lawrence,  Mass. 

1912  Sears,  Roebuck  &  Co Dallas,  Texas 

1912  Vineburg  Bldg Montreal,  Quebec 

1912  Imperial  Tobacco  Co Montreal,  Quebec 

1912  Richards  Pinhorn  Bldg Denver,  Col. 

1912  Kinney  &  Levan  Co.  Bldg Cleveland,  Ohio 

1912  Standard  Oil  Co.  Bldgs ...  (2) Cleveland,  Ohio 

1912  Silver  Sunshine  Bldgs ...  (2) Cleveland,  Ohio 

1912  Commercial  Improvement  Go's.  Bldg Columbus,  O. 

1912  Moore  Department  Store  Bldg Memphis,  Tenn. 

1912  Main  Eng.  Bldg.  Univ.  of  Minn Minneapolis,  Minn. 

1912  Honeyman  Hardware  Bldg Portland,  Ore. 

1912  Revillon  Wholesale  Hardware  Bldg Edmonton,  Alta. 

1912  Calgary  Furniture  Go's.  Bldg Calgary,  Alta. 

1912  Willoughby  Sumner  Bldg Saskatoon,  Sask. 

1912  U.  S.  Post  Office Minneapolis,  Minn. 

1912  Motor  Mart  Bldg Sioux  City,  la. 

1912  Finch  Van  Slyke  &  McConville  Bldg St.  Paul,  Minn. 

1912  Hudson  Bay  Go's.  Warehouse Winnipeg,  Man. 

1912  Snell  Bldg Moose  Jaw,  Sask. 

1913  Y.  M.  C.  A.  Bldg Vancouver,  B.  C. 

1913  Reynolds  Tobacco  Factory  Bldg Winston  Salem,  N.  C. 

1913  Ford  Motor  Bldg Memphis,  Tenn. 

1913  Ford  Motor  Bldg Los  Angeles,  Cal. 

1913  G.  Sommers  &  Co.  Bldg St.  Paul,  Minn. 

1913  Knickerbocker  Bldg Los  Angeles,  Cal. 

1913  Trinity  Auditorium  Bldg Los  Angeles,  Gal. 

1913  U.  S.  Alumium  Go's.  Bldg Pittsburg,  Pa. 

1913  Gordon  Fergusen  Go's.  Bldg St.  Paul,  Minn. 

1913  S.  H.  Kress  &  Go's.  Bldg Houston,  Tex. 

1913  "Los  Muchachos"  Bldg San  Juan,  Porto  Rico 


REC'D  L 

MAR    8* 


YC   13727 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


